Introduction and main results
In this paper, we consider entire solutions of the homogeneous Fisher–KPP equation
u t = Δ u + f ( u ) , for ( t , x ) ∈ R × R N , (1) u_t=\Delta u+f(u),\qquad\text{for }(t,x)\in\mathbb{R}\times\mathbb{R}^{N}, \tag*{(1)} u t = Δ u + f ( u ) , for ( t , x ) ∈ R × R N , (1) with N ≥ 1 N\ge1 N ≥ 1
f ∈ C 2 ( [ 0 , 1 ] ) , f ( 0 ) = f ( 1 ) = 0 , f ( u ) > 0 for 0 < u < 1 , f\in C^2([0,1]),\qquad f(0)=f(1)=0,\qquad f(u)>0\quad\text{for }0<u<1, f ∈ C 2 ([ 0 , 1 ]) , f ( 0 ) = f ( 1 ) = 0 , f ( u ) > 0 for 0 < u < 1 , f ′ ′ ( u ) ≤ 0 for 0 ≤ u ≤ 1. (2) f''(u)\le0\quad\text{for }0\le u\le1. \tag*{(2)} f ′′ ( u ) ≤ 0 for 0 ≤ u ≤ 1. (2) For convenience, we extend f f f C 1 C^1 C 1 R \mathbb{R} R f ( u ) = f ′ ( 0 ) u f(u)=f'(0)u f ( u ) = f ′ ( 0 ) u u < 0 u<0 u < 0 f ( u ) = f ′ ( 1 ) ( u − 1 ) f(u)=f'(1)(u-1) f ( u ) = f ′ ( 1 ) ( u − 1 ) u > 1 u>1 u > 1
f ′ ( 0 ) > 0 , f ′ ( 1 ) < 0 , f ( u ) ≤ f ′ ( 0 ) u for 0 ≤ u ≤ 1. f'(0)>0,\qquad f'(1)<0,\qquad f(u)\le f'(0)u\quad\text{for }0\le u\le1. f ′ ( 0 ) > 0 , f ′ ( 1 ) < 0 , f ( u ) ≤ f ′ ( 0 ) u for 0 ≤ u ≤ 1. A typical example is the logistic nonlinearity, that is, f ( u ) = u ( 1 − u ) f(u)=u(1-u) f ( u ) = u ( 1 − u ) u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] [16 ] and Kolmogorov, Petrovskii, and Piskunov [33 ] . Equation (1) also arises in models of population dynamics and other biological phenomena; see [2 , 3 , 15 , 16 , 17 , 38 , 44 , 52 ] . An entire solution is a classical solution defined for t ∈ R t\in\mathbb{R} t ∈ R x ∈ R N x\in\mathbb{R}^{N} x ∈ R N E \mathcal{E} E (2) , that is, every entire solution between 0 and 1 can be approximated by Hamel–Nadirashvili solutions generated by traveling fronts and the spatially homogeneous solution.
A typical entire solution of (1) is the traveling wave solution. It is well known that, in dimension N = 1 N=1 N = 1 (1) admits traveling wave solutions u ( t , x ) = ϕ c ( x + c t ) u(t,x)=\phi_c(x+ct) u ( t , x ) = ϕ c ( x + c t ) c ≥ c ∗ = 2 f ′ ( 0 ) c\ge c_* = 2\sqrt{f'(0)} c ≥ c ∗ = 2 f ′ ( 0 ) c ∗ c_* c ∗ [2 , 9 , 15 , 17 , 21 , 32 , 44 , 52 ] . The profile ϕ c \phi_c ϕ c c c c
ϕ ′ ′ − c ϕ ′ + f ( ϕ ) = 0 , ϕ ( − ∞ ) = 0 , ϕ ( + ∞ ) = 1 , ϕ ′ > 0. (3) \phi''-c\phi'+f(\phi)=0,\qquad\phi(-\infty)=0,\qquad\phi(+\infty)=1,\qquad\phi'>0. \tag*{(3)} ϕ ′′ − c ϕ ′ + f ( ϕ ) = 0 , ϕ ( − ∞ ) = 0 , ϕ ( + ∞ ) = 1 , ϕ ′ > 0. (3) The profile ϕ c \phi_c ϕ c ϕ c \phi_c ϕ c ϕ c ( x + c t ) \phi_c(x+ct) ϕ c ( x + c t ) ϕ c ( s ) \phi_c(s) ϕ c ( s ) s → − ∞ s\to-\infty s → − ∞ ϕ ′ ′ − c ϕ ′ + f ′ ( 0 ) ϕ = 0 \phi''-c\phi'+f'(0)\phi=0 ϕ ′′ − c ϕ ′ + f ′ ( 0 ) ϕ = 0
λ ( c ) = c − c 2 − 4 f ′ ( 0 ) 2 > 0 , λ + ( c ) = c + c 2 − 4 f ′ ( 0 ) 2 > 0. \lambda(c)=\frac{c-\sqrt{c^2-4f'(0)}}{2}>0,\qquad\lambda^+(c)=\frac{c+\sqrt{c^2-4f'(0)}}{2}>0. λ ( c ) = 2 c − c 2 − 4 f ′ ( 0 ) > 0 , λ + ( c ) = 2 c + c 2 − 4 f ′ ( 0 ) > 0. In particular, λ ∗ : = λ ( c ∗ ) = λ + ( c ∗ ) = c ∗ / 2 = f ′ ( 0 ) \lambda_*:=\lambda(c_*)=\lambda^+(c_*)=c_*/2=\sqrt{f'(0)} λ ∗ := λ ( c ∗ ) = λ + ( c ∗ ) = c ∗ /2 = f ′ ( 0 ) λ 2 − c ∗ λ + f ′ ( 0 ) = 0 \lambda^2-c_*\lambda+f'(0)=0 λ 2 − c ∗ λ + f ′ ( 0 ) = 0 ϕ c \phi_c ϕ c
for c > c ∗ , ϕ c ( s ) ∼ e λ ( c ) s as s → − ∞ , (4) \text{for }c>c_*,\qquad\phi_c(s)\sim e^{\lambda(c)s}\quad\text{as }s\to-\infty, \tag*{(4)} for c > c ∗ , ϕ c ( s ) ∼ e λ ( c ) s as s → − ∞ , (4) and
ϕ ∗ ( s ) : = ϕ c ∗ ( s ) ∼ ∣ s ∣ e λ ∗ s as s → − ∞ . (5) \phi_*(s):=\phi_{c_*}(s)\sim|s|e^{\lambda_*s}\quad\text{as }s\to-\infty. \tag*{(5)} ϕ ∗ ( s ) := ϕ c ∗ ( s ) ∼ ∣ s ∣ e λ ∗ s as s → − ∞. (5) More precisely, there exist B ∈ R B\in\mathbb{R} B ∈ R δ > 0 \delta>0 δ > 0
ϕ ∗ ( s ) = e λ ∗ s ( − s + B + O ( e δ s ) ) as s → − ∞ . (6) \phi_*(s)=e^{\lambda_*s}\left(-s+B+O(e^{\delta s})\right)\quad\text{as }s\to-\infty. \tag*{(6)} ϕ ∗ ( s ) = e λ ∗ s ( − s + B + O ( e δ s ) ) as s → − ∞. (6) The standard asymptotic estimates (4) and (5) can be found in [1 , 7 , 9 , 13 , 21 , 31 , 43 , 54 ] . We also use the logarithmic derivative estimates
0 < ϕ c ′ ( s ) ϕ c ( s ) < λ ( c ) for s ∈ R , ϕ c ′ ( s ) ϕ c ( s ) → λ ( c ) as s → − ∞ , c ≥ c ∗ . (7) 0 < \frac{\phi_c'(s)}{\phi_c(s)} < \lambda(c) \text{ for } s \in\mathbb{R}, \qquad\frac{\phi_c'(s)}{\phi_c(s)} \to\lambda(c) \text{ as } s \to-\infty,\qquad c \ge c_*. \tag*{(7)} 0 < ϕ c ( s ) ϕ c ′ ( s ) < λ ( c ) for s ∈ R , ϕ c ( s ) ϕ c ′ ( s ) → λ ( c ) as s → − ∞ , c ≥ c ∗ . (7) The estimates (7) follow from [25 ] . In higher dimensions, the corresponding planar traveling fronts are ϕ c ( x ⋅ ν + c t ) \phi_c(x \cdot\nu+ ct) ϕ c ( x ⋅ ν + c t ) ν ∈ S N − 1 \nu\in\mathbb{S}^{N-1} ν ∈ S N − 1 (4) and (5) .
In past decades, the theory of traveling wave solutions has been highly developed. When the coefficients of the equation depend on space and/or time, translation invariance is lost and such standard traveling fronts generally cease to exist, but new analogous notions have been introduced. In spatially and/or temporally periodic heterogeneous media, the natural analogue is a pulsating traveling front: the front propagates with a well-defined average speed while its profile is periodically modulated by the medium. Existence, uniqueness, stability, and spreading-speed properties of such fronts for KPP-type equations have been extensively studied, see [4 , 8 , 14 , 23 , 34 , 35 , 39 , 42 , 51 , 55 ] . This framework has also been extended to almost-periodic, almost-automorphic, recurrent, random, and uniquely ergodic settings, see [29 , 36 , 40 , 45 , 46 , 47 , 48 , 50 , 49 ] . For still more general nonperiodic or nonautonomous heterogeneous media, the notions of generalized transition front provide a flexible description of entire solutions connecting the invaded and uninvaded states through an interface of uniformly bounded width, without requiring a rigid profile, a constant instantaneous speed, or periodicity, see [5 , 6 , 22 , 24 , 26 , 58 ] .
Entire solutions describe complete trajectories of the parabolic dynamics, connecting prescribed asymptotic patterns in the past to their evolution. After a suitable space-time rescaling, they may also arise as local limiting profiles near regions where several interfaces interact. A natural strategy of constructing entire solutions is taking one or several traveling fronts as their asymptotic building blocks. For bistable equations, where f f f f ( 0 ) = f ( 1 ) = f ( θ ) = 0 f(0) = f(1) = f(\theta) = 0 f ( 0 ) = f ( 1 ) = f ( θ ) = 0 f < 0 f < 0 f < 0 ( 0 , θ ) (0,\theta) ( 0 , θ ) f > 0 f > 0 f > 0 ( θ , 1 ) (\theta,1) ( θ , 1 ) θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) [10 , 11 , 12 , 18 , 57 ] . Recent higher-dimensional results mix finitely many planar fronts to obtain transition fronts with changing shape, or obtain lower-dimensional and polyhedral entire solutions as limits of multidimensional traveling fronts [19 , 41 , 53 ] .
Hamel–Nadirashvili conjecture
We first recall the work of Hamel and Nadirashvili and their conjecture [27 , 28 ] . In the one-dimensional setting of (1) , they [27 ] used two basic building blocks to construct entire solutions. The first is the spatially homogeneous solution ξ f ( t ) \xi_f(t) ξ f ( t )
ξ f ′ ( t ) = f ( ξ f ( t ) ) , 0 < ξ f ( t ) < 1 , ξ f ( t ) ∼ e f ′ ( 0 ) t as t → − ∞ . (8) \xi_f'(t) = f(\xi_f(t)), \qquad0 < \xi_f(t) < 1, \qquad\xi_f(t) \sim e^{f'(0)t} \text{ as } t \to-\infty. \tag*{(8)} ξ f ′ ( t ) = f ( ξ f ( t )) , 0 < ξ f ( t ) < 1 , ξ f ( t ) ∼ e f ′ ( 0 ) t as t → − ∞. (8) This normalization determines ξ f \xi_f ξ f t → + ∞ t \to+\infty t → + ∞ [28 ] . For K > 0 K > 0 K > 0 K K K ξ f ( t + ln K / f ′ ( 0 ) ) \xi_f(t + \ln K/f'(0)) ξ f ( t + ln K / f ′ ( 0 ))
u c , h + ( t , x ) = ϕ c ( x + c t + h ) , u c ′ , h ′ − ( t , x ) = ϕ c ′ ( − x + c ′ t + h ′ ) , u^+_{c,h}(t,x) = \phi_c(x + ct + h), \qquad u^-_{c',h'}(t,x) = \phi_{c'}(-x + c't + h'), u c , h + ( t , x ) = ϕ c ( x + c t + h ) , u c ′ , h ′ − ( t , x ) = ϕ c ′ ( − x + c ′ t + h ′ ) , where c , c ′ > c ∗ c,c' > c_* c , c ′ > c ∗ ( c , h ) (c,h) ( c , h ) ( c ′ , h ′ ) (c',h') ( c ′ , h ′ ) ξ f \xi_f ξ f
In arbitrary dimensions, they [28 ] extended this construction by mixing planar fronts through suitable Radon measures. More precisely, let
X : = ( S N − 1 × [ c ∗ , + ∞ ) ) ∪ { ∞ } . X := \left(S^{N-1} \times[c_{*},+\infty)\right) \cup\{\infty\}. X := ( S N − 1 × [ c ∗ , + ∞ ) ) ∪ { ∞ } . The set S N − 1 × [ c ∗ , + ∞ ) S^{N-1} \times[c_{*},+\infty) S N − 1 × [ c ∗ , + ∞ ) A ⊂ X A \subset X A ⊂ X ∞ \infty ∞ ∞ ∈ A \infty\in A ∞ ∈ A
S N − 1 × [ c 0 , + ∞ ) ⊂ A S^{N-1} \times[c_{0},+\infty) \subset A S N − 1 × [ c 0 , + ∞ ) ⊂ A for some c 0 ≥ c ∗ c_{0} \ge c_{*} c 0 ≥ c ∗ X X X ∞ \infty ∞ M \mathcal{M} M ϑ \vartheta ϑ X X X ϑ \vartheta ϑ
ϑ ∣ S N − 1 × { c ∗ } = ∑ i = 1 k m i δ ( ν i , c ∗ ) , \left.\vartheta\right|_{S^{N-1}\times\{c_{*}\}}=\sum_{i=1}^{k}m_i\delta_{(\nu_i,c_{*})}, ϑ ∣ S N − 1 × { c ∗ } = i = 1 ∑ k m i δ ( ν i , c ∗ ) , where k ≥ 0 k \ge0 k ≥ 0 m i > 0 m_i > 0 m i > 0 ν i ∈ S N − 1 \nu_i \in S^{N-1} ν i ∈ S N − 1 ϑ ∈ M \vartheta\in\mathcal{M} ϑ ∈ M u ϑ ( t , x ) u_{\vartheta}(t,x) u ϑ ( t , x ) (1) which can be viewed as a mixture of a measurable set of traveling fronts. Moreover, the map
ϑ ↦ u ϑ \vartheta\mapsto u_{\vartheta} ϑ ↦ u ϑ is one-to-one on M \mathcal{M} M
M ^ : = { ϑ ∈ M : ϑ ( S N − 1 × { c ∗ } ) = 0 } \widehat{\mathcal{M}}:=\{\vartheta\in\mathcal{M}:\vartheta(S^{N-1}\times\{c_{*}\})=0\} M := { ϑ ∈ M : ϑ ( S N − 1 × { c ∗ }) = 0 } is continuous in the topology of [28 Theorem 1.2] . Their construction yields an infinite-dimensional family of entire solutions. Finite sums of Dirac masses recover finite mixtures of planar fronts, whereas more general measures produce multidimensional solutions, including nonplanar traveling fronts and radially symmetric entire solutions. They also proved the following partial uniqueness result. Let u ( t , x ) u(t,x) u ( t , x ) 0 < u < 1 0<u<1 0 < u < 1 ε > 0 \varepsilon>0 ε > 0
max ∣ x ∣ ≤ ( c ∗ + ε ) ∣ t ∣ u ( t , x ) ⟶ 0 as t → − ∞ , (9) \max_{|x|\le(c_{*}+\varepsilon)|t|}u(t,x)\longrightarrow0 \qquad\text{as }t\to-\infty, \tag*{(9)} ∣ x ∣ ≤ ( c ∗ + ε ) ∣ t ∣ max u ( t , x ) ⟶ 0 as t → − ∞ , (9) then there exists a unique measure ϑ ∈ M \vartheta\in\mathcal{M} ϑ ∈ M
u ( t , x ) = u ϑ ( t , x ) for all ( t , x ) ∈ R × R N , u(t,x)=u_{\vartheta}(t,x)\qquad\text{for all }(t,x)\in\mathbb{R}\times\mathbb{R}^{N}, u ( t , x ) = u ϑ ( t , x ) for all ( t , x ) ∈ R × R N , and supp ϑ ⊂ S N − 1 × [ c ∗ + ε , + ∞ ) ∪ { ∞ } \operatorname{supp}\vartheta\subset S^{N-1}\times[c_{*}+\varepsilon,+\infty)\cup\{\infty\} supp ϑ ⊂ S N − 1 × [ c ∗ + ε , + ∞ ) ∪ { ∞ }
Finally, they conjectured that the full class E \mathcal{E} E 0 ≤ u ≤ 1 0\le u\le1 0 ≤ u ≤ 1
E = { u ϑ : ϑ ∈ M } T , (10) \mathcal{E}=\left\{u_{\vartheta}:\vartheta\in\mathcal{M}\right\}^{\mathcal{T}}, \tag*{(10)} E = { u ϑ : ϑ ∈ M } T , (10) where T \mathcal{T} T C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2 E \mathcal{E} E T \mathcal{T} T (1) , analogous to the exponential functions in Widder’s representation of nonnegative entire solutions of the linear heat equation ∂ t v = Δ v \partial_t v=\Delta v ∂ t v = Δ v R N × R \mathbb{R}^{N}\times\mathbb{R} R N × R [56 ] and Lim and Zhang [37 ] for a Martin boundary formulation and extensions to complete noncompact manifolds.
The closure theorem
We prove (10) under the concavity assumptions (2) . In fact, every entire solution can be approximated in C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^{N}) C loc 1 , 2 ( R × R N ) m ≥ 0 m \ge0 m ≥ 0
v i ( t , x ) = ϕ c i ( x ⋅ ν i + c i t + h i ) , i = 1 , … , m , v_i(t,x)=\phi_{c_i}(x\cdot\nu_i+c_i t+h_i),\qquad i=1,\ldots,m, v i ( t , x ) = ϕ c i ( x ⋅ ν i + c i t + h i ) , i = 1 , … , m , where ν i ∈ S N − 1 \nu_i \in\mathbb{S}^{N-1} ν i ∈ S N − 1 c i ≥ c ∗ c_i \ge c_* c i ≥ c ∗ h i ∈ R h_i \in\mathbb{R} h i ∈ R
v 0 ( t ) = ξ f ( t + h 0 ) , h 0 ∈ R . v_0(t)=\xi_f(t+h_0),\qquad h_0 \in\mathbb{R}. v 0 ( t ) = ξ f ( t + h 0 ) , h 0 ∈ R . At least one component must be present: if m = 0 m=0 m = 0 v 0 v_0 v 0 m ≥ 1 m \ge1 m ≥ 1 ( c i , ν i ) (c_i,\nu_i) ( c i , ν i ) i = 1 , … , m i=1,\ldots,m i = 1 , … , m [28 Theorem 1.1] , there exists an entire solution u u u
max i v i ( t , x ) ≤ u ( t , x ) ≤ min { ∑ i v i ( t , x ) , 1 } for ( t , x ) ∈ R × R N . (11) \max_i v_i(t,x) \le u(t,x) \le\min\left\{\sum_i v_i(t,x),1\right\}\qquad\text{for }(t,x)\in\mathbb{R}\times\mathbb{R}^{N}. \tag*{(11)} i max v i ( t , x ) ≤ u ( t , x ) ≤ min { i ∑ v i ( t , x ) , 1 } for ( t , x ) ∈ R × R N . (11) We call this the finite HN solution . Uniqueness of the solution within these bounds is established in Theorem 3.4 . The identification of our finite approximants with these HN solutions, including their wave phases, is given in the proof of Theorem 1.1 . Equivalently, these finite HN solutions are the solutions u ϑ u_{\vartheta} u ϑ X X X [28 ] .
Theorem 1.1. For every entire solution u u u (1) with 0 ≤ u ≤ 1 0 \le u \le1 0 ≤ u ≤ 1
ϑ k = ∑ i = 1 m k a k , i δ ( ν k , i , c k , i ) + a k , 0 δ ∞ ∈ M , k = 1 , 2 , … , \vartheta_k=\sum_{i=1}^{m_k}a_{k,i}\delta_{(\nu_{k,i},c_{k,i})}+a_{k,0}\delta_{\infty}\in\mathcal{M},\qquad k=1,2,\ldots, ϑ k = i = 1 ∑ m k a k , i δ ( ν k , i , c k , i ) + a k , 0 δ ∞ ∈ M , k = 1 , 2 , … , where m k ≥ 0 m_k \ge0 m k ≥ 0 a k , 0 ≥ 0 a_{k,0} \ge0 a k , 0 ≥ 0
a k , i > 0 , ν k , i ∈ S N − 1 , c k , i ∈ [ c ∗ , + ∞ ) for i = 1 , … , m k , a_{k,i}>0,\qquad\nu_{k,i}\in\mathbb{S}^{N-1},\qquad c_{k,i}\in[c_*,+\infty)\qquad\text{for }i=1,\ldots,m_k, a k , i > 0 , ν k , i ∈ S N − 1 , c k , i ∈ [ c ∗ , + ∞ ) for i = 1 , … , m k , such that
u ϑ k → u in C l o c 1 , 2 ( R × R N ) as k → ∞ . u_{\vartheta_k}\to u\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^{N})\quad\text{as }k\to\infty. u ϑ k → u in C loc 1 , 2 ( R × R N ) as k → ∞. The term a k , 0 δ ∞ a_{k,0}\delta_{\infty} a k , 0 δ ∞ 0 0 0 1 1 1 ξ f \xi_f ξ f (2) .
Entire solutions from positive Borel measures on S N − 1 × { c ∗ } \mathbb{S}^{N-1}\times\{c_*\} S N − 1 × { c ∗ }
A key step in the proof of Theorem 1.1 is to construct and identify the critical part of an entire solution. This leads to a further result of independent interest: we remove the finite atomic restriction in the construction of Hamel and Nadirashvili [27 , 28 ] on the critical set S N − 1 × { c ∗ } \mathbb{S}^{N-1}\times\{c_*\} S N − 1 × { c ∗ } S N − 1 \mathbb{S}^{N-1} S N − 1 X X X
Let μ \mu μ S N − 1 \mathbb{S}^{N-1} S N − 1 M : = μ ( S N − 1 ) > 0 M:=\mu(\mathbb{S}^{N-1})>0 M := μ ( S N − 1 ) > 0
L μ ( x ) : = ∫ S N − 1 e λ ∗ x ⋅ ν d μ ( ν ) , z μ ( t , x ) : = c ∗ t + λ ∗ − 1 ln L μ ( x ) . (12) L_\mu(x):=\int_{\mathbb{S}^{N-1}}e^{\lambda_* x\cdot\nu}\,d\mu(\nu),\qquad z_\mu(t,x):=c_*t+\lambda_*^{-1}\ln L_\mu(x). \tag*{(12)} L μ ( x ) := ∫ S N − 1 e λ ∗ x ⋅ ν d μ ( ν ) , z μ ( t , x ) := c ∗ t + λ ∗ − 1 ln L μ ( x ) . (12) Multiplying μ \mu μ z μ z_\mu z μ μ \mu μ
P μ ( t , x ) : = 1 M ∫ S N − 1 ϕ ∗ ( x ⋅ ν + c ∗ t + ln M λ ∗ ) d μ ( ν ) . P_\mu(t,x):=\frac{1}{M}\int_{\mathbb{S}^{N-1}}\phi_*\left(x\cdot\nu+c_*t+\frac{\ln M}{\lambda_*}\right)\,d\mu(\nu). P μ ( t , x ) := M 1 ∫ S N − 1 ϕ ∗ ( x ⋅ ν + c ∗ t + λ ∗ ln M ) d μ ( ν ) . Since d μ / M d\mu/M d μ / M f f f
( P μ ) t − Δ P μ = 1 M ∫ S N − 1 f ( ϕ ∗ ( x ⋅ ν + c ∗ t + ln M λ ∗ ) ) d μ ( ν ) ≤ f ( P μ ) , (P_\mu)_t-\Delta P_\mu=\frac{1}{M}\int_{\mathbb{S}^{N-1}}f\left(\phi_*\left(x\cdot\nu+c_*t+\frac{\ln M}{\lambda_*}\right)\right)\,d\mu(\nu)\le f(P_\mu), ( P μ ) t − Δ P μ = M 1 ∫ S N − 1 f ( ϕ ∗ ( x ⋅ ν + c ∗ t + λ ∗ ln M ) ) d μ ( ν ) ≤ f ( P μ ) , so P μ P_\mu P μ
Q μ ( t , x ) : = 1 M ∫ S N − 1 e λ ∗ ( x ⋅ ν + c ∗ t + λ ∗ − 1 ln M ) d μ ( ν ) = e λ ∗ c ∗ t L μ ( x ) = e λ ∗ z μ ( t , x ) , Q_\mu(t,x):=\frac{1}{M}\int_{\mathbb{S}^{N-1}}e^{\lambda_*(x\cdot\nu+c_*t+\lambda_*^{-1}\ln M)}\,d\mu(\nu)=e^{\lambda_*c_*t}L_\mu(x)=e^{\lambda_*z_\mu(t,x)}, Q μ ( t , x ) := M 1 ∫ S N − 1 e λ ∗ ( x ⋅ ν + c ∗ t + λ ∗ − 1 l n M ) d μ ( ν ) = e λ ∗ c ∗ t L μ ( x ) = e λ ∗ z μ ( t , x ) , satisfies ( Q μ ) t − Δ Q μ = f ′ ( 0 ) Q μ (Q_\mu)_t-\Delta Q_\mu=f'(0)Q_\mu ( Q μ ) t − Δ Q μ = f ′ ( 0 ) Q μ f ( q ) ≤ f ′ ( 0 ) q f(q)\le f'(0)q f ( q ) ≤ f ′ ( 0 ) q 0 ≤ q ≤ 1 0\le q\le1 0 ≤ q ≤ 1 min { 1 , Q μ } \min\{1,Q_\mu\} min { 1 , Q μ } (1) . One might therefore try to construct an entire solution between P μ P_\mu P μ min { 1 , Q μ } \min\{1,Q_\mu\} min { 1 , Q μ } x = 0 x=0 x = 0
P μ ( t , 0 ) = ϕ ∗ ( c ∗ t + ln M λ ∗ ) , Q μ ( t , 0 ) = e λ ∗ c ∗ t + ln M , P_\mu(t,0)=\phi_*\left(c_*t+\frac{\ln M}{\lambda_*}\right),\qquad Q_\mu(t,0)=e^{\lambda_*c_*t+\ln M}, P μ ( t , 0 ) = ϕ ∗ ( c ∗ t + λ ∗ ln M ) , Q μ ( t , 0 ) = e λ ∗ c ∗ t + l n M , and (5) yields
P μ ( t , 0 ) Q μ ( t , 0 ) ∼ c ∗ ∣ t ∣ → + ∞ as t → − ∞ . \frac{P_\mu(t,0)}{Q_\mu(t,0)}\sim c_*|t|\to+\infty\qquad\text{as }t\to-\infty. Q μ ( t , 0 ) P μ ( t , 0 ) ∼ c ∗ ∣ t ∣ → + ∞ as t → − ∞. Thus Q μ Q_\mu Q μ P μ P_\mu P μ ϕ ∗ \phi_* ϕ ∗ ϕ ∗ ( z μ ) \phi_*(z_\mu) ϕ ∗ ( z μ )
The following theorem gives the critical-sphere construction and the injectivity and continuity of its measure parametrization.
Theorem 1.2. For every finite positive Borel measure μ \mu μ S N − 1 \mathbb{S}^{N-1} S N − 1 u μ u_\mu u μ (1) with ∂ t u μ > 0 \partial_tu_\mu>0 ∂ t u μ > 0
0 < ϕ ∗ ( z μ ( t , x ) ) ≤ u μ ( t , x ) < 1 for all ( t , x ) ∈ R × R N . (13) 0<\phi_*(z_\mu(t,x))\le u_\mu(t,x)<1\qquad\text{for all }(t,x)\in\mathbb{R}\times\mathbb{R}^N. \tag*{(13)} 0 < ϕ ∗ ( z μ ( t , x )) ≤ u μ ( t , x ) < 1 for all ( t , x ) ∈ R × R N . (13) Moreover,
\begin{aligned}
u_\mu(t,\cdot)&\to0 &&\text{as }t\to-\infty,\\
u_\mu(t,\cdot)&\to1 &&\text{as }t\to+\infty,
\end{aligned}
\qquad\text{in }C^2_{\mathrm{loc}}(\mathbb{R}^N). \tag*{(14)}\end{aligned}
For every α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) C α > 0 C_\alpha>0 C α > 0
{ x ∈ R N : u μ ( t , x ) = α } ⊂ { x ∈ R N : ∣ z μ ( t , x ) ∣ ≤ C α } for every t ∈ R . (15) \{x\in\mathbb{R}^N:u_\mu(t,x)=\alpha\}\subset\{x\in\mathbb{R}^N:|z_\mu(t,x)|\le C_\alpha\}\qquad\text{for every }t\in\mathbb{R}. \tag*{(15)} { x ∈ R N : u μ ( t , x ) = α } ⊂ { x ∈ R N : ∣ z μ ( t , x ) ∣ ≤ C α } for every t ∈ R . (15) Furthermore, the map μ ↦ u μ \mu\mapsto u_\mu μ ↦ u μ 1 S N − 1 \mathbb{S}^{N-1} S N − 1 C l o c 2 ( R × R N ) C^2_{\mathrm{loc}}(\mathbb{R}\times\mathbb{R}^N) C loc 2 ( R × R N )
The solution u μ u_\mu u μ ϕ ∗ ( z μ ) \phi_*(z_\mu) ϕ ∗ ( z μ ) (15) follows. This phase-strip inclusion does not, by itself, imply a uniform O ( 1 ) O(1) O ( 1 ) u μ u_\mu u μ z μ z_\mu z μ ∣ ∇ z μ ∣ |\nabla z_\mu| ∣∇ z μ ∣ u μ ≤ u ν u_\mu\leq u_\nu u μ ≤ u ν μ ≤ ν \mu\leq\nu μ ≤ ν
Constructing this family alone does not classify all entire solutions outside the partial uniqueness regime (9) . In fact, the failure of (9) does not force a solution to be generated solely by a measure on the critical sphere. An example in [20 ] is generated by a finite positive measure ϑ \vartheta ϑ X X X ϑ ( S N − 1 × { c ∗ } ) = 0 \vartheta(\mathbb{S}^{N-1} \times\{c_*\}) = 0 ϑ ( S N − 1 × { c ∗ }) = 0 (9) . Thus the critical-sphere construction complements Theorem 1.1 , whose approximation statement applies to all entire solutions in E \mathcal{E} E
Remark 1.3. The closure can also be described using mixtures of the original HN solutions with no critical-sphere mass and the critical solutions of Theorem 1.2 . With the convention u 0 = 0 u_0 = 0 u 0 = 0
E = { u ϑ ⊕ u μ : ϑ ∈ M ^ ∪ { 0 } , μ is a finite nonnegative Borel measure on S N − 1 } T . \mathcal{E} = \left\{ u_\vartheta\oplus u_\mu: \begin{array}{l} \vartheta\in\widehat{\mathcal{M}} \cup\{0\}, \\ \mu\text{ is a finite nonnegative Borel measure on } \mathbb{S}^{N-1} \end{array} \right\}^{\mathcal{T}}. E = { u ϑ ⊕ u μ : ϑ ∈ M ∪ { 0 } , μ is a finite nonnegative Borel measure on S N − 1 } T . Here ⊕ \oplus ⊕ § 6 . The condition ϑ ∈ M ^ \vartheta\in\widehat{\mathcal{M}} ϑ ∈ M ϑ ( S N − 1 × { c ∗ } ) = 0 \vartheta(\mathbb{S}^{N-1} \times\{c_*\}) = 0 ϑ ( S N − 1 × { c ∗ }) = 0 E \mathcal{E} E Theorem 1.1 . The equality therefore follows from Theorem 1.1 and the closedness of E \mathcal{E} E T \mathcal{T} T
Notations and organization
To compare nonlinear solutions of (1) with solutions of the linearized equation, we introduce the following notations. For a bounded measurable initial function h : R N → R h : \mathbb{R}^N \to\mathbb{R} h : R N → R v v v w w w v t = Δ v v_t = \Delta v v t = Δ v w t = Δ w + f ′ ( 0 ) w w_t = \Delta w + f^{\prime}(0)w w t = Δ w + f ′ ( 0 ) w t > 0 t > 0 t > 0 x ∈ R N x \in\mathbb{R}^N x ∈ R N v ( 0 , x ) = w ( 0 , x ) = h ( x ) v(0,x) = w(0,x) = h(x) v ( 0 , x ) = w ( 0 , x ) = h ( x )
( P t [ h ] ) ( x ) : = v ( t , x ) , ( L t [ h ] ) ( x ) : = w ( t , x ) for t ≥ 0 , x ∈ R N . (P_t[h])(x) := v(t,x), \qquad(L_t[h])(x) := w(t,x) \qquad\text{for } t \geq0,\ x \in\mathbb{R}^N. ( P t [ h ]) ( x ) := v ( t , x ) , ( L t [ h ]) ( x ) := w ( t , x ) for t ≥ 0 , x ∈ R N . Then P t [ h ] = e t Δ h P_t[h] = e^{t\Delta}h P t [ h ] = e t Δ h L t [ h ] = e f ′ ( 0 ) t P t [ h ] L_t[h] = e^{f^{\prime}(0)t}P_t[h] L t [ h ] = e f ′ ( 0 ) t P t [ h ] h h h ( S t [ h ] ) ( x ) : = u ( t , x ) (S_t[h])(x) := u(t,x) ( S t [ h ]) ( x ) := u ( t , x ) t ≥ 0 t \geq0 t ≥ 0 x ∈ R N x \in\mathbb{R}^N x ∈ R N u u u
{ u t = Δ u + f ( u ) , t > 0 , x ∈ R N , u ( 0 , x ) = h ( x ) , x ∈ R N . (16) \begin{cases}
u_t = \Delta u + f(u), & t > 0,\ x \in\mathbb{R}^N,\\
u(0,x) = h(x), & x \in\mathbb{R}^N. \tag*{(16)}
\end{cases} { u t = Δ u + f ( u ) , u ( 0 , x ) = h ( x ) , t > 0 , x ∈ R N , x ∈ R N . (16) Here f f f (2) . The nonlinear semigroup satisfies S 0 [ h ] = h S_0[h] = h S 0 [ h ] = h S t + s [ h ] = S t [ S s [ h ] ] S_{t+s}[h] = S_t[S_s[h]] S t + s [ h ] = S t [ S s [ h ]] t , s ≥ 0 t,s \geq0 t , s ≥ 0 L t [ h ] L_t[h] L t [ h ] S t [ h ] S_t[h] S t [ h ] t t t h h h
We also recall the representation used below: every nonnegative entire solution H H H H t = Δ H + f ′ ( 0 ) H H_t = \Delta H + f^{\prime}(0)H H t = Δ H + f ′ ( 0 ) H
H ( t , x ) = ∫ R N e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) , μ ( R N ) = H ( 0 , 0 ) , (17) H(t,x) = \int_{\mathbb{R}^N} e^{\lambda_* x \cdot\xi+ f^{\prime}(0)(1+|\xi|^2)t}\,d\mu(\xi), \qquad\mu(\mathbb{R}^N) = H(0,0), \tag*{(17)} H ( t , x ) = ∫ R N e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) , μ ( R N ) = H ( 0 , 0 ) , (17) where μ \mu μ e − f ′ ( 0 ) t H e^{-f^{\prime}(0)t}H e − f ′ ( 0 ) t H [37 Theorem 1.1] .
We organize the paper as follows. The differential estimates in Subsection 2.1 allow us to define nonlinear dilation D a D_a D a a > 0 a > 0 a > 0 ⊕ \oplus ⊕ − ∞ -\infty − ∞ ⊖ \ominus ⊖ 0 ≤ U ≤ V < 1 0 \le U \le V < 1 0 ≤ U ≤ V < 1 V ⊖ U V \ominus U V ⊖ U U ⊕ ( V ⊖ U ) = V U \oplus(V \ominus U) = V U ⊕ ( V ⊖ U ) = V U U U E U E_U E U U U U E U E_U E U
Building on these estimates and nonlinear operations, § 7 associates a solution of the linearized equation u t = Δ u + f ′ ( 0 ) u u_t = \Delta u + f'(0)u u t = Δ u + f ′ ( 0 ) u U U U (1) satisfying 0 < U < 1 0 < U < 1 0 < U < 1
sup T > 0 ( L T [ U ( − T , ⋅ ) ] ) ( 0 ) < ∞ . (18) \sup_{T>0} \left(L_T[U(-T,\cdot)]\right)(0) < \infty. \tag*{(18)} T > 0 sup ( L T [ U ( − T , ⋅ )] ) ( 0 ) < ∞. (18) We take U ( − T , ⋅ ) U(-T,\cdot) U ( − T , ⋅ ) − T -T − T 0 0 0 T → ∞ T \to\infty T → ∞ H H H H t = Δ H + f ′ ( 0 ) H H_t = \Delta H + f'(0)H H t = Δ H + f ′ ( 0 ) H
H ( t , ⋅ ) = lim T → ∞ L t + T [ U ( − T , ⋅ ) ] . H(t,\cdot) = \lim_{T\to\infty} L_{t+T}[U(-T,\cdot)]. H ( t , ⋅ ) = T → ∞ lim L t + T [ U ( − T , ⋅ )] . Moreover, the original nonlinear solution U U U H H H
U ( t , ⋅ ) = lim T → ∞ S t + T [ min { H ( − T , ⋅ ) , 1 } ] . U(t,\cdot) = \lim_{T\to\infty} S_{t+T}[\min\{H(-T,\cdot),1\}]. U ( t , ⋅ ) = T → ∞ lim S t + T [ min { H ( − T , ⋅ ) , 1 }] . Thus H H H U U U H H H U U U H N H_N H N
Small-amplitude analysis in the same section shows that the strict logarithmic gradient bound
∣ ∇ ln U ∣ ≤ λ ∗ r on R × R N , for some 0 ≤ r < 1 , (19) |\nabla\ln U| \le\lambda_* r \quad\text{on } \mathbb{R} \times\mathbb{R}^N,\quad\text{for some } 0 \le r < 1, \tag*{(19)} ∣∇ ln U ∣ ≤ λ ∗ r on R × R N , for some 0 ≤ r < 1 , (19) is sufficient for (18) . Under this bound, we prove
D a U a ⟶ H as a → 0 , \frac{D_a U}{a} \longrightarrow H \quad\text{as } a \to0, a D a U ⟶ H as a → 0 , where H H H
§ 8 then determines the relation between the time and space translation parameters. In the representation constructed in § 6 , translation by ( s , h ) (s,h) ( s , h ) e s λ + λ ∗ h ⋅ ξ e^{s\lambda+\lambda_* h\cdot\xi} e s λ + λ ∗ h ⋅ ξ
λ = f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) . \lambda= f'(0)(1+|\xi|^2). λ = f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) . Using this relation, we decompose an arbitrary entire solution into a critical part corresponding to ∣ ξ ∣ = 1 |\xi|=1 ∣ ξ ∣ = 1 (19) .
To identify the critical part, § 9 constructs a radially symmetric solution and uses it to build critical solutions from finite measures on the sphere. This section proves Theorem 1.2 , including the injectivity and continuity of the measure parametrization. We then show that the critical part obtained in § 8 belongs to this constructed family. For a nonzero critical solution U ∗ U_* U ∗ D a U ∗ / [ a ln ( 1 / a ) ] D_aU_* / [a\ln(1/a)] D a U ∗ / [ a ln ( 1/ a )] U ∗ U_* U ∗ § 10 completes the proof of Theorem 1.1 . We first fix a solution with a strict logarithmic gradient bound in the decomposition and approximate both its representing measure and the critical measure by finite atomic measures. We prove convergence of the corresponding nonlinear superpositions and identify them as finite H N H_N H N C l o c 1 , 2 ( R × R N ) C^{1,2}_{\mathrm{loc}}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N )
The ordered structure of the space of entire solutions
For a homogeneous linear equation, scalar multiples, sums, and differences of solutions are again solutions. These pointwise operations do not, in general, preserve solutions of the nonlinear equation (1) . We first establish differential estimates for entire solutions and then use Cauchy evolution to construct nonlinear analogues of these operations on the class of entire solutions satisfying 0 ≤ U < 1 0 \le U < 1 0 ≤ U < 1 D a U D_aU D a U a > 0 a > 0 a > 0 U ⊕ V U \oplus V U ⊕ V U ≤ V U \le V U ≤ V W W W U ⊕ W = V U \oplus W = V U ⊕ W = V E E E U U U E U E_U E U U U U C l o c 1 , 2 ( R × R N ) C^{1,2}_{\mathrm{loc}}(\mathbb{R} \times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) E U E_U E U
Differential estimates
Let U U U (1) with 0 < U < 1 0 < U < 1 0 < U < 1 R × R N \mathbb{R} \times\mathbb{R}^N R × R N ln U \ln U ln U 0 < u ≤ 1 0 < u \le1 0 < u ≤ 1 f f f
f ′ ( 0 ) u − f ( u ) = ∫ 0 u ( u − v ) ( − f ′ ′ ( v ) ) d v ≥ 0 , f ( u ) − u f ′ ( u ) = ∫ 0 u v ( − f ′ ′ ( v ) ) d v ≥ 0. (20) \begin{aligned}
f'(0)u-f(u)&=\int_0^u (u-v)(-f''(v))\,\mathrm{d}v \ge0,\\
f(u)-uf'(u)&=\int_0^u v(-f''(v))\,\mathrm{d}v \ge0. \tag*{(20)}
\end{aligned} f ′ ( 0 ) u − f ( u ) f ( u ) − u f ′ ( u ) = ∫ 0 u ( u − v ) ( − f ′′ ( v )) d v ≥ 0 , = ∫ 0 u v ( − f ′′ ( v )) d v ≥ 0. (20) In particular, 0 ≤ f ( u ) / u ≤ f ′ ( 0 ) 0 \le f(u)/u \le f'(0) 0 ≤ f ( u ) / u ≤ f ′ ( 0 ) f ′ ( u ) ≤ f ( u ) / u f'(u) \le f(u)/u f ′ ( u ) ≤ f ( u ) / u − f ′ ′ -f'' − f ′′
f ′ ( u ) = f ( u ) u ⟺ f ( v ) = f ′ ( 0 ) v for all 0 ≤ v ≤ u . (21) f'(u)=\frac{f(u)}{u}\quad\Longleftrightarrow\quad f(v)=f'(0)v\ \text{for all }0 \le v \le u. \tag*{(21)} f ′ ( u ) = u f ( u ) ⟺ f ( v ) = f ′ ( 0 ) v for all 0 ≤ v ≤ u . (21) Consequently f ( u ) / u < f ′ ( 0 ) f(u)/u < f'(0) f ( u ) / u < f ′ ( 0 ) f ′ ( u ) < f ( u ) / u f'(u) < f(u)/u f ′ ( u ) < f ( u ) / u
The following estimates control the spatial and temporal variation of ln U \ln U ln U
Theorem 2.1. Every classical entire solution U U U (1) with 0 < U < 1 0 < U < 1 0 < U < 1 R × R N \mathbb{R} \times\mathbb{R}^N R × R N
∣ ∇ ln U ∣ 2 ≤ ∂ t ln U , ∣ ∇ ln U ∣ ≤ λ ∗ , 0 < ∂ t ln U ≤ 2 f ′ ( 0 ) . (22) |\nabla\ln U|^2 \le\partial_t \ln U,\qquad|\nabla\ln U| \le\lambda_*,\qquad0 < \partial_t \ln U \le2f'(0). \tag*{(22)} ∣∇ ln U ∣ 2 ≤ ∂ t ln U , ∣∇ ln U ∣ ≤ λ ∗ , 0 < ∂ t ln U ≤ 2 f ′ ( 0 ) . (22) If ∣ ∇ ln U ∣ ≤ λ ∗ ℓ |\nabla\ln U| \le\lambda_*\ell ∣∇ ln U ∣ ≤ λ ∗ ℓ 0 ≤ ℓ ≤ 1 0 \le\ell\le1 0 ≤ ℓ ≤ 1 ∂ t ln U ≤ f ′ ( 0 ) ( 1 + ℓ 2 ) \partial_t \ln U \le f'(0)(1+\ell^2) ∂ t ln U ≤ f ′ ( 0 ) ( 1 + ℓ 2 ) and
0 < U t ≤ ( 1 + ℓ 2 ) f ( U ) . (23) 0 < U_t \le(1+\ell^2)f(U). \tag*{(23)} 0 < U t ≤ ( 1 + ℓ 2 ) f ( U ) . (23) In particular,
0 < U t ≤ 2 f ( U ) . (24) 0 < U_t \le2f(U). \tag*{(24)} 0 < U t ≤ 2 f ( U ) . (24) Moreover, U ( t , ⋅ ) → 0 U(t,\cdot) \to0 U ( t , ⋅ ) → 0 t → − ∞ t \to-\infty t → − ∞ U ( t , ⋅ ) → 1 U(t,\cdot) \to1 U ( t , ⋅ ) → 1 t → + ∞ t \to+\infty t → + ∞ R N \mathbb{R}^N R N
Proof. For convenience, we write
h ( t , x ) = ln U , p ( t , x ) = ∣ ∇ h ∣ 2 , r ( t , x ) = h t , L = ∂ t − Δ − 2 ∇ h ⋅ ∇ . h(t,x)=\ln U,\qquad p(t,x)=|\nabla h|^2,\qquad r(t,x)=h_t,\qquad\mathcal{L}=\partial_t-\Delta-2\nabla h\cdot\nabla. h ( t , x ) = ln U , p ( t , x ) = ∣∇ h ∣ 2 , r ( t , x ) = h t , L = ∂ t − Δ − 2∇ h ⋅ ∇. For any t 1 < t 2 t_1<t_2 t 1 < t 2 R > 0 R>0 R > 0 x 0 ∈ R N x_0\in\mathbb{R}^N x 0 ∈ R N U U U [ t 1 , t 2 ] × B R ( x 0 ) ‾ [t_1,t_2]\times\overline{B_R(x_0)} [ t 1 , t 2 ] × B R ( x 0 ) R × R N \mathbb{R}\times\mathbb{R}^N R × R N
h t = Δ h + p + f ( U ) U , (25) h_t=\Delta h+p+\frac{f(U)}{U}, \tag*{(25)} h t = Δ h + p + U f ( U ) , (25) and
L p = 2 ( f ′ ( U ) − f ( U ) U ) p − 2 ∣ ∇ 2 h ∣ 2 , L r = ( f ′ ( U ) − f ( U ) U ) r . (26) \mathcal{L}p=2\left(f'(U)-\frac{f(U)}{U}\right)p-2|\nabla^2h|^2,\qquad\mathcal{L}r=\left(f'(U)-\frac{f(U)}{U}\right)r. \tag*{(26)} L p = 2 ( f ′ ( U ) − U f ( U ) ) p − 2∣ ∇ 2 h ∣ 2 , L r = ( f ′ ( U ) − U f ( U ) ) r . (26) Here, ∣ ∇ 2 h ∣ |\nabla^2h| ∣ ∇ 2 h ∣
Step 1: the inequality p ≤ r p\le r p ≤ r Fix α > 1 \alpha>1 α > 1 F ( t , x ) = p ( t , x ) − α r ( t , x ) F(t,x)=p(t,x)-\alpha r(t,x) F ( t , x ) = p ( t , x ) − α r ( t , x ) (26) gives
L F = ( f ′ ( U ) − f ( U ) U ) ( 2 p − α r ) − 2 ∣ ∇ 2 h ∣ 2 = ( f ′ ( U ) − f ( U ) U ) ( p + F ) − 2 ∣ ∇ 2 h ∣ 2 . \begin{aligned}
\mathcal{L}F&=\left(f'(U)-\frac{f(U)}{U}\right)(2p-\alpha r)-2|\nabla^2h|^2\\
&=\left(f'(U)-\frac{f(U)}{U}\right)(p+F)-2|\nabla^2h|^2.
\end{aligned} L F = ( f ′ ( U ) − U f ( U ) ) ( 2 p − α r ) − 2∣ ∇ 2 h ∣ 2 = ( f ′ ( U ) − U f ( U ) ) ( p + F ) − 2∣ ∇ 2 h ∣ 2 . Also, (25) yields
Δ h = r − p − f ( U ) U = − F + ( α − 1 ) p + α f ( U ) / U α . \Delta h=r-p-\frac{f(U)}{U}=-\frac{F+(\alpha-1)p+\alpha f(U)/U}{\alpha}. Δ h = r − p − U f ( U ) = − α F + ( α − 1 ) p + α f ( U ) / U . On { F > 0 } \{F>0\} { F > 0 }
∣ ∇ 2 h ∣ 2 ≥ ( Δ h ) 2 N ≥ ( F + ( α − 1 ) p ) 2 N α 2 , |\nabla^2h|^2\ge\frac{(\Delta h)^2}{N}\ge\frac{(F+(\alpha-1)p)^2}{N\alpha^2}, ∣ ∇ 2 h ∣ 2 ≥ N ( Δ h ) 2 ≥ N α 2 ( F + ( α − 1 ) p ) 2 , and we obtain
L F ≤ − 2 N α 2 ( F + ( α − 1 ) p ) 2 . (27) \mathcal{L}F\le-\frac{2}{N\alpha^2}(F+(\alpha-1)p)^2. \tag*{(27)} L F ≤ − N α 2 2 ( F + ( α − 1 ) p ) 2 . (27) We construct a supersolution for F F F F F F 2 ∇ h 2\nabla h 2∇ h T 0 < t 1 T_0<t_1 T 0 < t 1 0 < ε < 1 0<\varepsilon<1 0 < ε < 1 U ε ( t , x ) U^\varepsilon(t,x) U ε ( t , x ) (1) on [ T 0 , t 1 ] × R N [T_0,t_1]\times\mathbb{R}^N [ T 0 , t 1 ] × R N
U ε ( T 0 , x ) = ε + ( 1 − ε ) U ( T 0 , x ) . (28) U^\varepsilon(T_0,x)=\varepsilon+(1-\varepsilon)U(T_0,x). \tag*{(28)} U ε ( T 0 , x ) = ε + ( 1 − ε ) U ( T 0 , x ) . (28) The comparison principle gives ε ≤ U ε ≤ 1 \varepsilon\le U^\varepsilon\le1 ε ≤ U ε ≤ 1 U ≤ U ε U\le U^\varepsilon U ≤ U ε
h ε = ln U ε , p ε = ∣ ∇ h ε ∣ 2 , r ε = ∂ t h ε , F ε = p ε − α r ε , h^\varepsilon=\ln U^\varepsilon,\qquad p^\varepsilon=|\nabla h^\varepsilon|^2,\qquad r^\varepsilon=\partial_t h^\varepsilon,\qquad F^\varepsilon=p^\varepsilon-\alpha r^\varepsilon, h ε = ln U ε , p ε = ∣∇ h ε ∣ 2 , r ε = ∂ t h ε , F ε = p ε − α r ε , and
L ε = ∂ t − Δ − 2 ∇ h ε ⋅ ∇ . \mathcal{L}^\varepsilon=\partial_t-\Delta-2\nabla h^\varepsilon\cdot\nabla. L ε = ∂ t − Δ − 2∇ h ε ⋅ ∇. Interior estimates for U U U U ε U^\varepsilon U ε Q = [ T 0 , t 1 ] × R N Q=[T_0,t_1]\times\mathbb{R}^N Q = [ T 0 , t 1 ] × R N
∥ ∇ U ε ∥ L ∞ ( Q ) + ∥ U t ε ∥ L ∞ ( Q ) ≤ K , (29) \|\nabla U^\varepsilon\|_{L^\infty(Q)}+\|U^\varepsilon_t\|_{L^\infty(Q)}\le K, \tag*{(29)} ∥∇ U ε ∥ L ∞ ( Q ) + ∥ U t ε ∥ L ∞ ( Q ) ≤ K , (29) with K K K ε \varepsilon ε U ε ≥ ε U^\varepsilon\ge\varepsilon U ε ≥ ε
∥ ∇ h ε ∥ L ∞ ( Q ) ≤ K ε , ∥ F ε ∥ L ∞ ( Q ) ≤ K 2 ε 2 + α K ε . \|\nabla h^\varepsilon\|_{L^\infty(Q)}\le\frac{K}{\varepsilon},\qquad\|F^\varepsilon\|_{L^\infty(Q)}\le\frac{K^2}{\varepsilon^2}+\frac{\alpha K}{\varepsilon}. ∥∇ h ε ∥ L ∞ ( Q ) ≤ ε K , ∥ F ε ∥ L ∞ ( Q ) ≤ ε 2 K 2 + ε α K . Thus the drift 2 ∇ h ε 2\nabla h^{\varepsilon} 2∇ h ε F ε F^{\varepsilon} F ε ε \varepsilon ε
The calculation for (27) applies to U ε U^{\varepsilon} U ε { F ε > 0 } \{F^{\varepsilon}>0\} { F ε > 0 }
L ε F ε ≤ − 2 N α 2 ( F ε + ( α − 1 ) p ε ) 2 ≤ − 2 N α 2 ( F ε ) 2 . \mathcal{L}^{\varepsilon}F^{\varepsilon}\leq-\frac{2}{N\alpha^{2}}\left(F^{\varepsilon}+(\alpha-1)p^{\varepsilon}\right)^{2}\leq-\frac{2}{N\alpha^{2}}\left(F^{\varepsilon}\right)^{2}. L ε F ε ≤ − N α 2 2 ( F ε + ( α − 1 ) p ε ) 2 ≤ − N α 2 2 ( F ε ) 2 . Choose
M ε = max { 1 , sup x ∈ R N F ε ( T 0 , x ) } , F ‾ ε ( t ) = N α 2 M ε N α 2 + 2 M ε ( t − T 0 ) . M_{\varepsilon}=\max\left\{1,\sup_{x\in\mathbb{R}^{N}}F^{\varepsilon}(T_{0},x)\right\},\qquad\overline{F}^{\varepsilon}(t)=\frac{N\alpha^{2}M_{\varepsilon}}{N\alpha^{2}+2M_{\varepsilon}(t-T_{0})}. M ε = max { 1 , x ∈ R N sup F ε ( T 0 , x ) } , F ε ( t ) = N α 2 + 2 M ε ( t − T 0 ) N α 2 M ε . This positive function F ‾ ε ( t ) \overline{F}^{\varepsilon}(t) F ε ( t )
( F ‾ ε ) ′ = − 2 N α 2 ( F ‾ ε ) 2 , F ε ( T 0 , x ) ≤ F ‾ ε ( T 0 ) . (\overline{F}^{\varepsilon})'=-\frac{2}{N\alpha^{2}}(\overline{F}^{\varepsilon})^{2},\qquad F^{\varepsilon}(T_{0},x)\leq\overline{F}^{\varepsilon}(T_{0}). ( F ε ) ′ = − N α 2 2 ( F ε ) 2 , F ε ( T 0 , x ) ≤ F ε ( T 0 ) . The comparison principle gives that for T 0 < t ≤ t 1 T_{0}<t\leq t_{1} T 0 < t ≤ t 1
F ε ( t , x ) ≤ F ‾ ε ( t ) = N α 2 N α 2 M ε − 1 + 2 ( t − T 0 ) ≤ N α 2 2 ( t − T 0 ) . (30) F^{\varepsilon}(t,x)\leq\overline{F}^{\varepsilon}(t)=\frac{N\alpha^{2}}{N\alpha^{2}M_{\varepsilon}^{-1}+2(t-T_{0})}\leq\frac{N\alpha^{2}}{2(t-T_{0})}. \tag*{(30)} F ε ( t , x ) ≤ F ε ( t ) = N α 2 M ε − 1 + 2 ( t − T 0 ) N α 2 ≤ 2 ( t − T 0 ) N α 2 . (30) The difference D ε = U ε − U D^{\varepsilon}=U^{\varepsilon}-U D ε = U ε − U D ε ≥ 0 D^{\varepsilon}\geq0 D ε ≥ 0
( ∂ t − Δ ) D ε = ( ∫ 0 1 f ′ ( U + θ D ε ) d θ ) D ε ≤ f ′ ( 0 ) D ε . (\partial_{t}-\Delta)D^{\varepsilon}=\left(\int_{0}^{1}f'(U+\theta D^{\varepsilon})\,d\theta\right)D^{\varepsilon}\leq f'(0)D^{\varepsilon}. ( ∂ t − Δ ) D ε = ( ∫ 0 1 f ′ ( U + θ D ε ) d θ ) D ε ≤ f ′ ( 0 ) D ε . Together with 0 ≤ D ε ( T 0 , x ) ≤ ε 0\leq D^{\varepsilon}(T_{0},x)\leq\varepsilon 0 ≤ D ε ( T 0 , x ) ≤ ε 0 ≤ D ε ( t , x ) ≤ ε e f ′ ( 0 ) ( t − T 0 ) 0\leq D^{\varepsilon}(t,x)\leq\varepsilon e^{f'(0)(t-T_{0})} 0 ≤ D ε ( t , x ) ≤ ε e f ′ ( 0 ) ( t − T 0 ) T 0 < t 1 T_{0}<t_{1} T 0 < t 1
( U ε , ∇ U ε , U t ε ) ( t 1 , ⋅ ) → ( U , ∇ U , U t ) ( t 1 , ⋅ ) in C l o c ( R N ) as ε → 0. (31) (U^{\varepsilon},\nabla U^{\varepsilon},U_{t}^{\varepsilon})(t_{1},\cdot)\to(U,\nabla U,U_{t})(t_{1},\cdot)\quad\text{in }C_{\mathrm{loc}}(\mathbb{R}^{N})\text{ as }\varepsilon\to0. \tag*{(31)} ( U ε , ∇ U ε , U t ε ) ( t 1 , ⋅ ) → ( U , ∇ U , U t ) ( t 1 , ⋅ ) in C loc ( R N ) as ε → 0. (31) Since U > 0 U>0 U > 0 F ε ( t 1 , x ) → F ( t 1 , x ) F^{\varepsilon}(t_{1},x)\to F(t_{1},x) F ε ( t 1 , x ) → F ( t 1 , x ) ε → 0 \varepsilon\to0 ε → 0 ε → 0 \varepsilon\to0 ε → 0 (30) gives
p ( t 1 , x ) − α r ( t 1 , x ) ≤ N α 2 2 ( t 1 − T 0 ) . p(t_{1},x)-\alpha r(t_{1},x)\leq\frac{N\alpha^{2}}{2(t_{1}-T_{0})}. p ( t 1 , x ) − α r ( t 1 , x ) ≤ 2 ( t 1 − T 0 ) N α 2 . Let T 0 → − ∞ T_{0}\to-\infty T 0 → − ∞ α → 1 \alpha\to1 α → 1 p ≤ r p\leq r p ≤ r p ≥ 0 p\geq0 p ≥ 0 U t = U r ≥ 0 U_{t}=U_{r}\geq0 U t = U r ≥ 0 U t U_{t} U t ( ∂ t − Δ ) U t = f ′ ( U ) U t (\partial_{t}-\Delta)U_{t}=f'(U)U_{t} ( ∂ t − Δ ) U t = f ′ ( U ) U t U t ≡ 0 U_{t}\equiv0 U t ≡ 0 U t > 0 U_{t}>0 U t > 0 U t ≡ 0 U_{t}\equiv0 U t ≡ 0 p ≤ r p\leq r p ≤ r ∇ U = 0 \nabla U=0 ∇ U = 0 U U U (1) would then give f ( U ) = 0 f(U)=0 f ( U ) = 0 0 < U < 1 0<U<1 0 < U < 1 U t > 0 U_{t}>0 U t > 0
Step 2: the bound p ≤ f ′ ( 0 ) p\leq f'(0) p ≤ f ′ ( 0 ) Fix T 0 < t 1 T_{0}<t_{1} T 0 < t 1 (28) and the notation of Step 1 on [ T 0 , t 1 ] × R N [T_{0},t_{1}]\times\mathbb{R}^{N} [ T 0 , t 1 ] × R N
Q ε = U ε ( ( t − T 0 ) p ε + h ε − f ′ ( 0 ) ( t − T 0 ) ) . Q^{\varepsilon}=U^{\varepsilon}\left((t-T_{0})p^{\varepsilon}+h^{\varepsilon}-f'(0)(t-T_{0})\right). Q ε = U ε ( ( t − T 0 ) p ε + h ε − f ′ ( 0 ) ( t − T 0 ) ) . Then Q ε ( T 0 , ⋅ ) = U ε h ε ≤ 0 Q^{\varepsilon}(T_{0},\cdot)=U^{\varepsilon}h^{\varepsilon}\leq0 Q ε ( T 0 , ⋅ ) = U ε h ε ≤ 0 f ( U ε ) / U ε ≤ f ′ ( 0 ) f(U^{\varepsilon})/U^{\varepsilon}\leq f'(0) f ( U ε ) / U ε ≤ f ′ ( 0 ) L ε h ε = f ( U ε ) / U ε − p ε \mathcal{L}^{\varepsilon}h^{\varepsilon}=f(U^{\varepsilon})/U^{\varepsilon}-p^{\varepsilon} L ε h ε = f ( U ε ) / U ε − p ε L ε p ε ≤ 0 \mathcal{L}^{\varepsilon}p^{\varepsilon}\leq0 L ε p ε ≤ 0
( ∂ t − Δ ) Q ε = f ( U ε ) U ε Q ε + U ε ( ( t − T 0 ) L ε p ε + f ( U ε ) U ε − f ′ ( 0 ) ) ≤ f ( U ε ) U ε Q ε . (\partial_{t}-\Delta)Q^{\varepsilon}=\frac{f(U^{\varepsilon})}{U^{\varepsilon}}Q^{\varepsilon}+U^{\varepsilon}\left((t-T_{0})\mathcal{L}^{\varepsilon}p^{\varepsilon}+\frac{f(U^{\varepsilon})}{U^{\varepsilon}}-f'(0)\right)\leq\frac{f(U^{\varepsilon})}{U^{\varepsilon}}Q^{\varepsilon}. ( ∂ t − Δ ) Q ε = U ε f ( U ε ) Q ε + U ε ( ( t − T 0 ) L ε p ε + U ε f ( U ε ) − f ′ ( 0 ) ) ≤ U ε f ( U ε ) Q ε . By (29) , ε ≤ U ε ≤ 1 \varepsilon\leq U^{\varepsilon}\leq1 ε ≤ U ε ≤ 1 h ε ≤ 0 h^{\varepsilon}\leq0 h ε ≤ 0
Q ε ≤ ( t 1 − T 0 ) K 2 ε on [ T 0 , t 1 ] × R N . Q^{\varepsilon}\leq\frac{(t_{1}-T_{0})K^{2}}{\varepsilon}\quad\text{on }[T_{0},t_{1}]\times\mathbb{R}^{N}. Q ε ≤ ε ( t 1 − T 0 ) K 2 on [ T 0 , t 1 ] × R N . The maximum principle therefore gives Q ε ≤ 0 Q^\varepsilon\le0 Q ε ≤ 0
( t − T 0 ) p ε ≤ − h ε + f ′ ( 0 ) ( t − T 0 ) . (t-T_0)p^\varepsilon\le-h^\varepsilon+f'(0)(t-T_0). ( t − T 0 ) p ε ≤ − h ε + f ′ ( 0 ) ( t − T 0 ) . At t = t 1 t=t_1 t = t 1 ε → 0 \varepsilon\to0 ε → 0 (31) and U ( t 1 , x ) > 0 U(t_1,x)>0 U ( t 1 , x ) > 0
p ( t 1 , x ) ≤ f ′ ( 0 ) − h ( t 1 , x ) t 1 − T 0 . p(t_1,x)\le f'(0)-\frac{h(t_1,x)}{t_1-T_0}. p ( t 1 , x ) ≤ f ′ ( 0 ) − t 1 − T 0 h ( t 1 , x ) . Letting T 0 → − ∞ T_0\to-\infty T 0 → − ∞ p ≤ f ′ ( 0 ) p\le f'(0) p ≤ f ′ ( 0 ) ∣ ∇ h ∣ ≤ λ ∗ |\nabla h|\le\lambda_* ∣∇ h ∣ ≤ λ ∗
Step 3: the bound r ≤ f ′ ( 0 ) ( 1 + ℓ 2 ) r\le f'(0)(1+\ell^2) r ≤ f ′ ( 0 ) ( 1 + ℓ 2 ) We first show that sup r \sup r sup r ( t ∗ , x ∗ ) (t_*,x_*) ( t ∗ , x ∗ )
V ( t , x ) = U ( t , x ) U ( t ∗ , x ∗ ) . V(t,x)=\frac{U(t,x)}{U(t_*,x_*)}. V ( t , x ) = U ( t ∗ , x ∗ ) U ( t , x ) . On ( t ∗ − 4 , t ∗ ] × B 2 ( x ∗ ) (t_*-4,t_*]\times B_2(x_*) ( t ∗ − 4 , t ∗ ] × B 2 ( x ∗ ) U t > 0 U_t>0 U t > 0 ∣ ∇ ln U ∣ ≤ λ ∗ |\nabla\ln U|\le\lambda_* ∣∇ ln U ∣ ≤ λ ∗
U ( t , x ) ≤ U ( t ∗ , x ) ≤ e λ ∗ ∣ x − x ∗ ∣ U ( t ∗ , x ∗ ) ≤ e 2 λ ∗ U ( t ∗ , x ∗ ) . U(t,x)\le U(t_*,x)\le e^{\lambda_*|x-x_*|}U(t_*,x_*)\le e^{2\lambda_*}U(t_*,x_*). U ( t , x ) ≤ U ( t ∗ , x ) ≤ e λ ∗ ∣ x − x ∗ ∣ U ( t ∗ , x ∗ ) ≤ e 2 λ ∗ U ( t ∗ , x ∗ ) . Thus, 0 < V ≤ e 2 λ ∗ 0<V\le e^{2\lambda_*} 0 < V ≤ e 2 λ ∗ ( t ∗ − 4 , t ∗ ] × B 2 ( x ∗ ) (t_*-4,t_*]\times B_2(x_*) ( t ∗ − 4 , t ∗ ] × B 2 ( x ∗ ) V V V V t − Δ V = ( f ( U ) / U ) V V_t-\Delta V=(f(U)/U)V V t − Δ V = ( f ( U ) / U ) V u ↦ f ( u ) / u u\mapsto f(u)/u u ↦ f ( u ) / u C 1 C^1 C 1 [ 0 , 1 ] [0,1] [ 0 , 1 ] U U U 0 < U < 1 0<U<1 0 < U < 1 f ( U ) / U f(U)/U f ( U ) / U V V V
∣ V t ( t ∗ , x ∗ ) ∣ + ∣ ∇ 2 V ( t ∗ , x ∗ ) ∣ ≤ C e 2 λ ∗ , |V_t(t_*,x_*)|+|\nabla^2V(t_*,x_*)|\le Ce^{2\lambda_*}, ∣ V t ( t ∗ , x ∗ ) ∣ + ∣ ∇ 2 V ( t ∗ , x ∗ ) ∣ ≤ C e 2 λ ∗ , where C C C ( t ∗ , x ∗ ) (t_*,x_*) ( t ∗ , x ∗ ) U ( t ∗ , x ∗ ) U(t_*,x_*) U ( t ∗ , x ∗ )
∣ U t ( t ∗ , x ∗ ) ∣ + ∣ ∇ 2 U ( t ∗ , x ∗ ) ∣ U ( t ∗ , x ∗ ) ≤ C e 2 λ ∗ . \frac{|U_t(t_*,x_*)|+|\nabla^2U(t_*,x_*)|}{U(t_*,x_*)}\le Ce^{2\lambda_*}. U ( t ∗ , x ∗ ) ∣ U t ( t ∗ , x ∗ ) ∣ + ∣ ∇ 2 U ( t ∗ , x ∗ ) ∣ ≤ C e 2 λ ∗ . Finally, ∇ 2 h = ∇ 2 U / U − ∇ h ⊗ ∇ h \nabla^2h=\nabla^2U/U-\nabla h\otimes\nabla h ∇ 2 h = ∇ 2 U / U − ∇ h ⊗ ∇ h ∣ ∇ h ∣ 2 ≤ f ′ ( 0 ) |\nabla h|^2\le f'(0) ∣∇ h ∣ 2 ≤ f ′ ( 0 )
∣ ∇ 2 h ∣ ≤ C e 2 λ ∗ + f ′ ( 0 ) < ∞ , 0 < r = h t ≤ C e 2 λ ∗ < ∞ . (32) |\nabla^2h|\le Ce^{2\lambda_*}+f'(0)<\infty,\qquad0<r=h_t\le Ce^{2\lambda_*}<\infty. \tag*{(32)} ∣ ∇ 2 h ∣ ≤ C e 2 λ ∗ + f ′ ( 0 ) < ∞ , 0 < r = h t ≤ C e 2 λ ∗ < ∞. (32) Assume ∣ ∇ h ∣ ≤ λ ∗ ℓ |\nabla h|\le\lambda_*\ell ∣∇ h ∣ ≤ λ ∗ ℓ 0 ≤ ℓ ≤ 1 0\le\ell\le1 0 ≤ ℓ ≤ 1 M : = sup R × R N r > f ′ ( 0 ) ( 1 + ℓ 2 ) M:=\sup_{\mathbb{R}\times\mathbb{R}^N}r>f'(0)(1+\ell^2) M := sup R × R N r > f ′ ( 0 ) ( 1 + ℓ 2 ) (32) , one has M ≤ C e 2 λ ∗ < ∞ M\le Ce^{2\lambda_*}<\infty M ≤ C e 2 λ ∗ < ∞ ( t n , x n ) (t_n,x_n) ( t n , x n ) r ( t n , x n ) → M r(t_n,x_n)\to M r ( t n , x n ) → M
w n ( t , x ) = U ( t + t n , x + x n ) U ( t n , x n ) . w_n(t,x)=\frac{U(t+t_n,x+x_n)}{U(t_n,x_n)}. w n ( t , x ) = U ( t n , x n ) U ( t + t n , x + x n ) . Then, w n ( 0 , 0 ) = 1 w_n(0,0)=1 w n ( 0 , 0 ) = 1 ∣ ∇ h ∣ ≤ λ ∗ ℓ |\nabla h|\le\lambda_*\ell ∣∇ h ∣ ≤ λ ∗ ℓ (32) that
∣ ln w n ( t , x ) ∣ = ∣ h ( t n + t , x n + x ) − h ( t n , x n ) ∣ ≤ λ ∗ ℓ ∣ x ∣ + C e 2 λ ∗ ∣ t ∣ . |\ln w_n(t,x)|=|h(t_n+t,x_n+x)-h(t_n,x_n)|
\le\lambda_*\ell|x|+Ce^{2\lambda_*}|t|. ∣ ln w n ( t , x ) ∣ = ∣ h ( t n + t , x n + x ) − h ( t n , x n ) ∣ ≤ λ ∗ ℓ ∣ x ∣ + C e 2 λ ∗ ∣ t ∣. It implies
e − λ ∗ ℓ ∣ x ∣ − R 0 ∣ t ∣ ≤ w n ( t , x ) ≤ e λ ∗ ℓ ∣ x ∣ + R 0 ∣ t ∣ . (33) e^{-\lambda_*\ell|x|-R_0|t|}\le w_n(t,x)\le e^{\lambda_*\ell|x|+R_0|t|}. \tag*{(33)} e − λ ∗ ℓ ∣ x ∣ − R 0 ∣ t ∣ ≤ w n ( t , x ) ≤ e λ ∗ ℓ ∣ x ∣ + R 0 ∣ t ∣ . (33) where R 0 : = C e 2 λ ∗ R_0:=Ce^{2\lambda_*} R 0 := C e 2 λ ∗ a n = U ( t n , x n ) a_n=U(t_n,x_n) a n = U ( t n , x n ) w n ( t , x ) w_n(t,x) w n ( t , x )
( w n ) t − Δ w n = f ( a n w n ) a n . (w_n)_t-\Delta w_n=\frac{f(a_nw_n)}{a_n}. ( w n ) t − Δ w n = a n f ( a n w n ) . By the bound (33) , 0 < a n < 1 0<a_n<1 0 < a n < 1 w n w_n w n w w w w ( 0 , 0 ) = 1 w(0,0)=1 w ( 0 , 0 ) = 1
w t = Δ w + F a ( w ) , F a ( v ) = { f ( a v ) / a , a > 0 , f ′ ( 0 ) v , a = 0. w_t=\Delta w+F_a(w),\qquad F_a(v)=
\begin{cases}
f(av)/a, & a>0,\\
f'(0)v, & a=0.
\end{cases} w t = Δ w + F a ( w ) , F a ( v ) = { f ( a v ) / a , f ′ ( 0 ) v , a > 0 , a = 0. Here a a a a n a_n a n a n w n < 1 a_nw_n<1 a n w n < 1 a w ≤ 1 aw\leq1 a w ≤ 1 a = 0 a=0 a = 0 f ( a n v ) / a n → f ′ ( 0 ) v f(a_nv)/a_n\to f'(0)v f ( a n v ) / a n → f ′ ( 0 ) v v v v h ~ = ln w \tilde{h}=\ln w h ~ = ln w r ~ = h ~ t \tilde{r}=\tilde{h}_t r ~ = h ~ t L ~ = ∂ t − Δ − 2 ∇ h ~ ⋅ ∇ \tilde{\mathcal{L}}=\partial_t-\Delta-2\nabla\tilde{h}\cdot\nabla L ~ = ∂ t − Δ − 2∇ h ~ ⋅ ∇
∣ ∇ h ~ ∣ ≤ λ ∗ ℓ , r ~ ≤ M , r ~ ( 0 , 0 ) = M , |\nabla\tilde{h}|\leq\lambda_*\ell,\qquad\tilde{r}\leq M,\qquad\tilde{r}(0,0)=M, ∣∇ h ~ ∣ ≤ λ ∗ ℓ , r ~ ≤ M , r ~ ( 0 , 0 ) = M , and
h ~ t = Δ h ~ + ∣ ∇ h ~ ∣ 2 + f ( a w ) a w , \tilde{h}_t=\Delta\tilde{h}+|\nabla\tilde{h}|^2+\frac{f(aw)}{aw}, h ~ t = Δ h ~ + ∣∇ h ~ ∣ 2 + a w f ( a w ) , L ~ r ~ = ( f ′ ( a w ) − f ( a w ) a w ) r ~ . \tilde{\mathcal{L}}\tilde{r}=\left(f'(aw)-\frac{f(aw)}{aw}\right)\tilde{r}. L ~ r ~ = ( f ′ ( a w ) − a w f ( a w ) ) r ~ . Clearly, M M M r ~ \tilde{r} r ~ r ~ ( t , x ) = M \tilde{r}(t,x)=M r ~ ( t , x ) = M ( − ∞ , 0 ] × R N (-\infty,0]\times\mathbb{R}^N ( − ∞ , 0 ] × R N t t t
Δ h ~ = M − ∣ ∇ h ~ ∣ 2 − f ( a w ) a w ≥ M − f ′ ( 0 ) ( 1 + ℓ 2 ) > 0. \Delta\tilde{h}=M-|\nabla\tilde{h}|^2-\frac{f(aw)}{aw}\geq M-f'(0)(1+\ell^2)>0. Δ h ~ = M − ∣∇ h ~ ∣ 2 − a w f ( a w ) ≥ M − f ′ ( 0 ) ( 1 + ℓ 2 ) > 0. This contradicts the gradient bound by the divergence theorem. More precisely, for every L > 0 L>0 L > 0 ν \nu ν
( M − f ′ ( 0 ) ( 1 + ℓ 2 ) ) ∣ B L ∣ ≤ ∫ B L Δ h ~ d x = ∫ ∂ B L ∇ h ~ ⋅ ν d S ≤ λ ∗ ℓ ∣ ∂ B L ∣ . \left(M-f'(0)(1+\ell^2)\right)|B_L|\leq\int_{B_L}\Delta\tilde{h}\,dx=\int_{\partial B_L}\nabla\tilde{h}\cdot\nu\,dS\leq\lambda_*\ell|\partial B_L|. ( M − f ′ ( 0 ) ( 1 + ℓ 2 ) ) ∣ B L ∣ ≤ ∫ B L Δ h ~ d x = ∫ ∂ B L ∇ h ~ ⋅ ν d S ≤ λ ∗ ℓ ∣ ∂ B L ∣. Since ∣ ∂ B L ∣ / ∣ B L ∣ = N / L |\partial B_L|/|B_L|=N/L ∣ ∂ B L ∣/∣ B L ∣ = N / L L → ∞ L\to\infty L → ∞ M > f ′ ( 0 ) ( 1 + ℓ 2 ) M>f'(0)(1+\ell^2) M > f ′ ( 0 ) ( 1 + ℓ 2 ) M ≤ f ′ ( 0 ) ( 1 + ℓ 2 ) M\leq f'(0)(1+\ell^2) M ≤ f ′ ( 0 ) ( 1 + ℓ 2 ) ℓ = 1 \ell=1 ℓ = 1 0 < r ≤ 2 f ′ ( 0 ) 0<r\leq2f'(0) 0 < r ≤ 2 f ′ ( 0 )
Step 4: the bound U t ≤ ( 1 + ℓ 2 ) f ( U ) U_t\leq(1+\ell^2)f(U) U t ≤ ( 1 + ℓ 2 ) f ( U ) Assume ∣ ∇ h ∣ ≤ λ ∗ ℓ |\nabla h|\leq\lambda_*\ell ∣∇ h ∣ ≤ λ ∗ ℓ 0 ≤ ℓ ≤ 1 0\leq\ell\leq1 0 ≤ ℓ ≤ 1 λ = 1 + ℓ 2 \lambda=1+\ell^2 λ = 1 + ℓ 2 0 < r ≤ f ′ ( 0 ) λ 0<r\leq f'(0)\lambda 0 < r ≤ f ′ ( 0 ) λ f ( U ) / U f(U)/U f ( U ) / U
L ( f ( U ) U ) = ( f ′ ( U ) − f ( U ) U ) f ( U ) U − U f ′ ′ ( U ) p . (34) \mathcal{L}\left(\frac{f(U)}{U}\right)=\left(f'(U)-\frac{f(U)}{U}\right)\frac{f(U)}{U}-Uf''(U)p. \tag*{(34)} L ( U f ( U ) ) = ( f ′ ( U ) − U f ( U ) ) U f ( U ) − U f ′′ ( U ) p . (34) For d = λ f ( U ) / U − r d=\lambda f(U)/U-r d = λ f ( U ) / U − r (26) and (34) imply
( L − f ′ ( U ) + f ( U ) U ) d = − λ U f ′ ′ ( U ) p ≥ 0. (35) \left(\mathcal{L}-f'(U)+\frac{f(U)}{U}\right)d=-\lambda Uf''(U)p\geq0. \tag*{(35)} ( L − f ′ ( U ) + U f ( U ) ) d = − λ U f ′′ ( U ) p ≥ 0. (35) Suppose that m = inf d < 0 m=\inf d<0 m = inf d < 0 r ≤ f ′ ( 0 ) λ r\leq f'(0)\lambda r ≤ f ′ ( 0 ) λ f ( u ) / u → f ′ ( 0 ) f(u)/u\to f'(0) f ( u ) / u → f ′ ( 0 ) u ↘ 0 u\searrow0 u ↘ 0 U ≥ a 0 > 0 U\geq a_0>0 U ≥ a 0 > 0 d d d m m m d = 0 d=0 d = 0 ( 0 , 1 ) (0,1) ( 0 , 1 ) λ f ( U ) / U − r = m < 0 \lambda f(U)/U-r=m<0 λ f ( U ) / U − r = m < 0 r ≤ f ′ ( 0 ) λ r\leq f'(0)\lambda r ≤ f ′ ( 0 ) λ f ( U ) / U < f ′ ( 0 ) f(U)/U<f'(0) f ( U ) / U < f ′ ( 0 ) f ′ ( U ) − f ( U ) / U < 0 f'(U)-f(U)/U<0 f ′ ( U ) − f ( U ) / U < 0 (21) . Formula (35) then gives
L d = ( f ′ ( U ) − f ( U ) U ) m − λ U f ′ ′ ( U ) p > 0. \mathcal{L}d=\left(f'(U)-\frac{f(U)}{U}\right)m-\lambda Uf''(U)p>0. L d = ( f ′ ( U ) − U f ( U ) ) m − λ U f ′′ ( U ) p > 0. At a space–time minimum the left side is nonpositive, a contradiction. Consequently r ≤ λ f ( U ) / U r\leq\lambda f(U)/U r ≤ λ f ( U ) / U (23) and (24) .
Finally, since U t > 0 U_t>0 U t > 0 0 < U < 1 0<U<1 0 < U < 1
U ± ( x ) : = lim t → ± ∞ U ( t , x ) U_\pm(x):=\lim_{t\to\pm\infty}U(t,x) U ± ( x ) := t → ± ∞ lim U ( t , x ) exist and satisfy
0 ≤ U − ( x ) ≤ U ( 0 , x ) ≤ U + ( x ) ≤ 1. 0\leq U_-(x)\leq U(0,x)\leq U_+(x)\leq1. 0 ≤ U − ( x ) ≤ U ( 0 , x ) ≤ U + ( x ) ≤ 1. For any sequence t n → ± ∞ t_n \to\pm\infty t n → ± ∞ U ( t + t n , x ) → U ± ( x ) U(t+t_n,x) \to U_\pm(x) U ( t + t n , x ) → U ± ( x ) C l o c 1 , 2 ( R × R N ) C^{1,2}_{\mathrm{loc}}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N )
Passing to the limit in the equation and in ∣ ∇ U ∣ 2 ≤ U U t |\nabla U|^2 \leq UU_t ∣∇ U ∣ 2 ≤ U U t
Δ U ± + f ( U ± ) = 0 , ∣ ∇ U ± ∣ 2 ≤ 0. \Delta U_\pm+ f(U_\pm) = 0,\qquad|\nabla U_\pm|^2 \leq0. Δ U ± + f ( U ± ) = 0 , ∣∇ U ± ∣ 2 ≤ 0. Thus U ± U_\pm U ± U − ( x ) ≤ U ( 0 , x ) < 1 U_-(x) \leq U(0,x) < 1 U − ( x ) ≤ U ( 0 , x ) < 1 U + ( x ) ≥ U ( 0 , x ) > 0 U_+(x) \geq U(0,x) > 0 U + ( x ) ≥ U ( 0 , x ) > 0 U − ≡ 0 U_- \equiv0 U − ≡ 0 U + ≡ 1 U_+ \equiv1 U + ≡ 1 U ( t , ⋅ ) → 0 U(t,\cdot) \to0 U ( t , ⋅ ) → 0 t → − ∞ t \to-\infty t → − ∞ U ( t , ⋅ ) → 1 U(t,\cdot) \to1 U ( t , ⋅ ) → 1 t → + ∞ t \to+\infty t → + ∞ R N \mathbb{R}^N R N
Nonlinear constructions and amplitude estimates
We construct dilation and superposition from the Cauchy evolution and establish the amplitude estimates needed for their algebraic laws and small-amplitude limits.
All Cauchy problems below use the concave extension fixed after (2) . Given bounded nonnegative initial data h T h_T h T − T -T − T S t + T [ h T ] S_{t+T}[h_T] S t + T [ h T ] t > − T t>-T t > − T pullback approximation . A limit of these solutions as T → ∞ T\to\infty T → ∞ t t t pullback limit . Since f ′ ( 1 ) < 0 f'(1)<0 f ′ ( 1 ) < 0 [ 0 , 1 ] [0,1] [ 0 , 1 ]
Let E \mathcal{E} E [ 0 , 1 ] [0,1] [ 0 , 1 ] C = { 0 } ∪ { U ∈ E : 0 < U < 1 } \mathcal{C}=\{0\}\cup\{U\in\mathcal{E}:0<U<1\} C = { 0 } ∪ { U ∈ E : 0 < U < 1 } U U U (1) with 0 < U < 1 0<U<1 0 < U < 1
∣ ∇ ln U ∣ ≤ λ ∗ , 0 < U t ≤ 2 f ( U ) ≤ 2 f ′ ( 0 ) U , U ( t , ⋅ ) → 0 ( t → − ∞ ) . (36) |\nabla\ln U|\leq\lambda_*,\qquad0<U_t\leq2f(U)\leq2f'(0)U,\qquad U(t,\cdot)\to0\quad(t\to-\infty). \tag*{(36)} ∣∇ ln U ∣ ≤ λ ∗ , 0 < U t ≤ 2 f ( U ) ≤ 2 f ′ ( 0 ) U , U ( t , ⋅ ) → 0 ( t → − ∞ ) . (36) All unqualified inequalities between solutions are pointwise.
We first record the comparison, concavity, and stability properties of the Cauchy operators that will control the nonlinear constructions.
Lemma 2.2. For bounded nonnegative initial data h ( x ) h(x) h ( x ) k ( x ) k(x) k ( x ) t ≥ 0 t\geq0 t ≥ 0
S t [ h + k ] ≤ S t [ h ] + S t [ k ] , S_t[h+k]\leq S_t[h]+S_t[k], S t [ h + k ] ≤ S t [ h ] + S t [ k ] , and
S t [ θ h + ( 1 − θ ) k ] ≥ θ S t [ h ] + ( 1 − θ ) S t [ k ] , for 0 ≤ θ ≤ 1. (37) S_t[\theta h+(1-\theta)k]\geq\theta S_t[h]+(1-\theta)S_t[k],\qquad\text{for }0\leq\theta\leq1. \tag*{(37)} S t [ θ h + ( 1 − θ ) k ] ≥ θ S t [ h ] + ( 1 − θ ) S t [ k ] , for 0 ≤ θ ≤ 1. (37) In particular,
S t [ a h ] ≥ a S t [ h ] , for 0 < a ≤ 1 , S t [ a h ] ≤ a S t [ h ] , for a ≥ 1. (38) S_t[ah]\geq aS_t[h],\qquad\text{for }0<a\leq1,\qquad S_t[ah]\leq aS_t[h],\qquad\text{for }a\geq1. \tag*{(38)} S t [ ah ] ≥ a S t [ h ] , for 0 < a ≤ 1 , S t [ ah ] ≤ a S t [ h ] , for a ≥ 1. (38) Moreover
∣ S t [ h ] − S t [ k ] ∣ ≤ S t [ ∣ h − k ∣ ] ≤ e f ′ ( 0 ) t e t Δ ∣ h − k ∣ . (39) |S_t[h]-S_t[k]|\leq S_t[|h-k|]\leq e^{f'(0)t}e^{t\Delta}|h-k|. \tag*{(39)} ∣ S t [ h ] − S t [ k ] ∣ ≤ S t [ ∣ h − k ∣ ] ≤ e f ′ ( 0 ) t e t Δ ∣ h − k ∣. (39) Proof. For u , v ≥ 0 u,v\geq0 u , v ≥ 0
f ( u ) + f ( v ) − f ( u + v ) = ∫ 0 u ( f ′ ( s ) − f ′ ( s + v ) ) d s ≥ 0 , f(u)+f(v)-f(u+v)=\int_0^u\left(f'(s)-f'(s+v)\right)\,\mathrm{d}s\geq0, f ( u ) + f ( v ) − f ( u + v ) = ∫ 0 u ( f ′ ( s ) − f ′ ( s + v ) ) d s ≥ 0 , so the sum of two solutions is a supersolution. On the other hand, concavity also gives
θ f ( u ) + ( 1 − θ ) f ( v ) − f ( θ u + ( 1 − θ ) v ) ≤ 0. \theta f(u)+(1-\theta)f(v)-f(\theta u+(1-\theta)v)\leq0. θ f ( u ) + ( 1 − θ ) f ( v ) − f ( θ u + ( 1 − θ ) v ) ≤ 0. The first two inequalities follow from the comparison principle. Since S t [ 0 ] = 0 S_t[0] = 0 S t [ 0 ] = 0 (37) to h h h 0 0 0 θ = a \theta= a θ = a 0 < a ≤ 1 0 < a \le1 0 < a ≤ 1 a h ah ah 0 0 0 θ = 1 / a \theta= 1/a θ = 1/ a a ≥ 1 a \ge1 a ≥ 1 u = S t [ h ] u = S_t[h] u = S t [ h ] v = S t [ k ] v = S_t[k] v = S t [ k ]
( u − v ) t − Δ ( u − v ) = f ( u ) − f ( v ) ≤ f ( u − v ) where u > v . (u-v)_t-\Delta(u-v)=f(u)-f(v)\le f(u-v) \qquad\text{where } u>v. ( u − v ) t − Δ ( u − v ) = f ( u ) − f ( v ) ≤ f ( u − v ) where u > v . Thus max { u − v , 0 } \max\{u-v,0\} max { u − v , 0 } max { h − k , 0 } \max\{h-k,0\} max { h − k , 0 } ∣ S t [ h ] − S t [ k ] ∣ ≤ S t [ ∣ h − k ∣ ] |S_t[h]-S_t[k]|\le S_t[|h-k|] ∣ S t [ h ] − S t [ k ] ∣ ≤ S t [ ∣ h − k ∣ ] (39) follows from f ( u ) ≤ f ′ ( 0 ) u f(u)\le f'(0)u f ( u ) ≤ f ′ ( 0 ) u u ≥ 0 u\ge0 u ≥ 0
For U , V ∈ C U,V\in\mathcal{C} U , V ∈ C a > 0 a>0 a > 0 D a U D_aU D a U U ⊕ V U\oplus V U ⊕ V
D a U ( t ) = lim T → ∞ S t + T [ a U ( − T ) ] , U ⊕ V = lim T → ∞ S t + T [ U ( − T ) + V ( − T ) ] . (40) D_aU(t)=\lim_{T\to\infty}S_{t+T}[aU(-T)],\qquad U\oplus V=\lim_{T\to\infty}S_{t+T}[U(-T)+V(-T)]. \tag*{(40)} D a U ( t ) = T → ∞ lim S t + T [ a U ( − T )] , U ⊕ V = T → ∞ lim S t + T [ U ( − T ) + V ( − T )] . (40) Here and below T > − t T>-t T > − t T T T a ≤ 1 a\le1 a ≤ 1 a ≥ 1 a\ge1 a ≥ 1 − T 2 -T_2 − T 2 − T 1 -T_1 − T 1 T 2 > T 1 T_2>T_1 T 2 > T 1 t t t C l o c 1 , 2 ( R × R N ) \mathcal{C}_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N )
a U ≤ D a U ≤ U for 0 < a ≤ 1 , U ≤ D a U ≤ a U for a ≥ 1 , aU\le D_aU\le U\qquad\text{for }0<a\le1,\qquad U\le D_aU\le aU\qquad\text{for }a\ge1, a U ≤ D a U ≤ U for 0 < a ≤ 1 , U ≤ D a U ≤ a U for a ≥ 1 , max ( U , V ) ≤ U ⊕ V ≤ min ( U + V , 1 ) . (41) \max(U,V)\le U\oplus V\le\min(U+V,1). \tag*{(41)} max ( U , V ) ≤ U ⊕ V ≤ min ( U + V , 1 ) . (41) The past limits and the strong maximum principle put all these solutions in C \mathcal{C} C U ⊕ V U\oplus V U ⊕ V U + V U+V U + V a ↦ D a U ( t , x ) a\mapsto D_aU(t,x) a ↦ D a U ( t , x )
The spatial bound proved above must also hold after changing the amplitude a a a
Lemma 2.3. Let U U U (1) with 0 < U < 1 0<U<1 0 < U < 1 ∣ ∇ ln U ∣ ≤ λ ∗ r |\nabla\ln U|\le\lambda_*r ∣∇ ln U ∣ ≤ λ ∗ r r ≥ 0 r\ge0 r ≥ 0 D a U D_aU D a U S t + T [ a U ( − T , ⋅ ) ] S_{t+T}[aU(-T,\cdot)] S t + T [ a U ( − T , ⋅ )] a > 0 a>0 a > 0 T > 0 T>0 T > 0 t > − T t>-T t > − T
Proof. Integrating the assumed bound along a line segment at time − T -T − T
U ( − T , x + y ) ≤ e λ ∗ r ∣ y ∣ U ( − T , x ) for x , y ∈ R N . U(-T,x+y)\le e^{\lambda_*r|y|}U(-T,x)\qquad\text{for }x,y\in\mathbb{R}^N. U ( − T , x + y ) ≤ e λ ∗ r ∣ y ∣ U ( − T , x ) for x , y ∈ R N . For a > 0 a>0 a > 0 t > − T t>-T t > − T (38) therefore yield
( S t + T [ a U ( − T , ⋅ ) ] ) ( x + y ) = ( S t + T [ a U ( − T , ⋅ + y ) ] ) ( x ) ≤ ( S t + T [ e λ ∗ r ∣ y ∣ a U ( − T , ⋅ ) ] ) ( x ) ≤ e λ ∗ r ∣ y ∣ ( S t + T [ a U ( − T , ⋅ ) ] ) ( x ) . \begin{aligned}
(S_{t+T}[aU(-T,\cdot)])(x+y)&=(S_{t+T}[aU(-T,\cdot+y)])(x)\\
&\le(S_{t+T}[e^{\lambda_*r|y|}aU(-T,\cdot)])(x)\\
&\le e^{\lambda_*r|y|}(S_{t+T}[aU(-T,\cdot)])(x).
\end{aligned} ( S t + T [ a U ( − T , ⋅ )]) ( x + y ) = ( S t + T [ a U ( − T , ⋅ + y )]) ( x ) ≤ ( S t + T [ e λ ∗ r ∣ y ∣ a U ( − T , ⋅ )]) ( x ) ≤ e λ ∗ r ∣ y ∣ ( S t + T [ a U ( − T , ⋅ )]) ( x ) . Taking logarithms and interchanging x x x x + y x+y x + y T → ∞ T\to\infty T → ∞ D a U D_aU D a U
We next determine the endpoint limits of the amplitude parameter a a a
Lemma 2.4. For every entire solution U U U (1) with 0 < U < 1 0<U<1 0 < U < 1
lim a → 0 D a U = 0 , lim a → ∞ D a U = 1 in C l o c 1 , 2 . \lim_{a\to0}D_aU=0,\qquad\lim_{a\to\infty}D_aU=1\quad\text{in }C^{1,2}_{\mathrm{loc}}. a → 0 lim D a U = 0 , a → ∞ lim D a U = 1 in C loc 1 , 2 . Proof. Integration of ∂ t ln U ≤ 2 f ′ ( 0 ) \partial_t\ln U\le2f'(0) ∂ t ln U ≤ 2 f ′ ( 0 ) t + ln a 2 f ′ ( 0 ) t+\frac{\ln a}{2f'(0)} t + 2 f ′ ( 0 ) l n a t t t a ≤ 1 a\le1 a ≤ 1 a U ( t , x ) ≤ U ( t + ln a 2 f ′ ( 0 ) , x ) aU(t,x)\le U\left(t+\frac{\ln a}{2f'(0)},x\right) a U ( t , x ) ≤ U ( t + 2 f ′ ( 0 ) l n a , x ) D a U ≤ U ( t + ln a 2 f ′ ( 0 ) , x ) D_aU\le U\left(t+\frac{\ln a}{2f'(0)},x\right) D a U ≤ U ( t + 2 f ′ ( 0 ) l n a , x ) a ≥ 1 a\ge1 a ≥ 1 D a U ≥ U ( t + ln a 2 f ′ ( 0 ) , x ) D_aU\ge U\left(t+\frac{\ln a}{2f'(0)},x\right) D a U ≥ U ( t + 2 f ′ ( 0 ) l n a , x ) Theorem 2.1 .
For the subsequent amplitude comparisons, define the critical-profile slope as a function of the solution value:
q ( u ) = ϕ ∗ ′ ( ϕ ∗ − 1 ( u ) ) for 0 < u < 1. q(u)=\phi_*'\left(\phi_*^{-1}(u)\right)\quad\text{for }0<u<1. q ( u ) = ϕ ∗ ′ ( ϕ ∗ − 1 ( u ) ) for 0 < u < 1. Then q q ′ = c ∗ q − f qq'=c_*q-f q q ′ = c ∗ q − f 0 < q ( u ) < λ ∗ u 0<q(u)<\lambda_*u 0 < q ( u ) < λ ∗ u q ( u ) / u → λ ∗ q(u)/u\to\lambda_* q ( u ) / u → λ ∗ lim u ↗ 1 q ( u ) = 0 \lim_{u\nearrow1}q(u)=0 lim u ↗ 1 q ( u ) = 0 (7) and the traveling-wave equation (3) .
The spatial gradient bound also yields a second-order differential inequality for the amplitude map a ↦ D a U ( t , x ) a\mapsto D_aU(t,x) a ↦ D a U ( t , x )
Lemma 2.5. If ∣ ∇ ln U ∣ ≤ λ ∗ r |\nabla\ln U|\le\lambda_*r ∣∇ ln U ∣ ≤ λ ∗ r 0 ≤ r ≤ 1 0\le r\le1 0 ≤ r ≤ 1 G ( a ) = D a U ( t , x ) G(a)=D_aU(t,x) G ( a ) = D a U ( t , x ) γ ∈ [ r 2 , 1 ] \gamma\in[r^2,1] γ ∈ [ r 2 , 1 ]
γ a 2 G ′ ′ ≤ a G ′ − f ( G ) f ′ ( 0 ) (42) \gamma a^2G''\le aG'-\frac{f(G)}{f'(0)} \tag*{(42)} γ a 2 G ′′ ≤ a G ′ − f ′ ( 0 ) f ( G ) (42) in the distributional sense on ( 0 , ∞ ) (0,\infty) ( 0 , ∞ )
Proof. The extended f f f C 1 , 1 ( R ) C^{1,1}(\mathbb{R}) C 1 , 1 ( R ) ρ ε \rho_\varepsilon ρ ε
f ε = ρ ε ∗ f − ( ρ ε ∗ f ) ( 0 ) . f_\varepsilon=\rho_\varepsilon*f-(\rho_\varepsilon*f)(0). f ε = ρ ε ∗ f − ( ρ ε ∗ f ) ( 0 ) . Then f ε → f f_\varepsilon\to f f ε → f C l o c 1 C^1_{\mathrm{loc}} C loc 1 f ε ( 0 ) = 0 f_\varepsilon(0)=0 f ε ( 0 ) = 0 f ε ′ ≤ f ′ ( 0 ) f_\varepsilon'\le f'(0) f ε ′ ≤ f ′ ( 0 ) f ε ′ ′ ≤ 0 f_\varepsilon''\le0 f ε ′′ ≤ 0 T T T u u u u t = Δ u + f ε ( u ) u_t=\Delta u+f_\varepsilon(u) u t = Δ u + f ε ( u ) u ( − T ) = a U ( − T ) u(-T)=aU(-T) u ( − T ) = a U ( − T ) v = a ∂ a u v=a\partial_au v = a ∂ a u w = a 2 ∂ a 2 u w=a^2\partial_a^2u w = a 2 ∂ a 2 u L u = ∂ t − Δ − f ε ′ ( u ) \mathcal{L}_u=\partial_t-\Delta-f_\varepsilon'(u) L u = ∂ t − Δ − f ε ′ ( u ) a a a
L u v = 0 , L u w = f ε ′ ′ ( u ) v 2 , L u f ε ( u ) = − f ε ′ ′ ( u ) ∣ ∇ u ∣ 2 . \mathcal{L}_uv=0,\qquad\mathcal{L}_uw=f_\varepsilon''(u)v^2,\qquad\mathcal{L}_uf_\varepsilon(u)=-f_\varepsilon''(u)|\nabla u|^2. L u v = 0 , L u w = f ε ′′ ( u ) v 2 , L u f ε ( u ) = − f ε ′′ ( u ) ∣∇ u ∣ 2 . Initially v = u v=u v = u w = 0 w=0 w = 0 u u u v v v ∣ ∇ u ∣ ≤ λ ∗ r v |\nabla u|\le\lambda_*rv ∣∇ u ∣ ≤ λ ∗ r v
L u { γ w + f ε ( u ) f ′ ( 0 ) − v } = f ε ′ ′ ( u ) ( γ v 2 − ∣ ∇ u ∣ 2 f ′ ( 0 ) ) ≤ 0. \mathcal{L}_u\left\{\gamma w+\frac{f_\varepsilon(u)}{f'(0)}-v\right\}
=f_\varepsilon''(u)\left(\gamma v^2-\frac{|\nabla u|^2}{f'(0)}\right)\le0. L u { γ w + f ′ ( 0 ) f ε ( u ) − v } = f ε ′′ ( u ) ( γ v 2 − f ′ ( 0 ) ∣∇ u ∣ 2 ) ≤ 0. Its initial value is f ε ( a U ) / f ′ ( 0 ) − a U ≤ 0 f_\varepsilon(aU)/f'(0)-aU\le0 f ε ( a U ) / f ′ ( 0 ) − a U ≤ 0 γ a 2 u a a ≤ a u a − f ε ( u ) / f ′ ( 0 ) \gamma a^2u_{aa}\le au_a-f_\varepsilon(u)/f'(0) γ a 2 u aa ≤ a u a − f ε ( u ) / f ′ ( 0 ) ψ ∈ C c ∞ ( ( 0 , ∞ ) ) \psi\in C^\infty_c((0,\infty)) ψ ∈ C c ∞ (( 0 , ∞ ))
∫ 0 ∞ ( γ u ( a 2 ψ ) ′ ′ + u ( a ψ ) ′ + f ε ( u ) f ′ ( 0 ) ψ ) d a ≤ 0. \int_0^\infty\left(\gamma u(a^2\psi)''+u(a\psi)'+\frac{f_\varepsilon(u)}{f'(0)}\psi\right)\,da\le0. ∫ 0 ∞ ( γ u ( a 2 ψ ) ′′ + u ( a ψ ) ′ + f ′ ( 0 ) f ε ( u ) ψ ) d a ≤ 0. For fixed T T T 0 ≤ u ( t ) ≤ a e f ′ ( 0 ) ( t + T ) ∥ U ( − T ) ∥ ∞ 0\le u(t)\le ae^{f'(0)(t+T)}\|U(-T)\|_\infty 0 ≤ u ( t ) ≤ a e f ′ ( 0 ) ( t + T ) ∥ U ( − T ) ∥ ∞ ε ↘ 0 \varepsilon\searrow0 ε ↘ 0 S t + T [ a U ( − T ) ] S_{t+T}[aU(-T)] S t + T [ a U ( − T )] f f f f ε f_\varepsilon f ε a a a G G G ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) T → ∞ T\to\infty T → ∞
The following proposition compares nonlinear dilation with a phase shift of the critical traveling front and gives its asymptotic behavior.
Proposition 2.6. Every entire solution U U U (1) with 0 < U < 1 0<U<1 0 < U < 1 R × R N \mathbb{R}\times\mathbb{R}^{N} R × R N ( t , x ) ∈ R × R N (t,x)\in\mathbb{R}\times\mathbb{R}^{N} ( t , x ) ∈ R × R N
D a U ( t , x ) ≤ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( U ( t , x ) ) ) for 0 < a ≤ 1 , (43) D_aU(t,x)\leq\phi_*\left(\lambda_*^{-1}\ln a+\phi_*^{-1}(U(t,x))\right)\qquad\text{for }0<a\leq1, \tag*{(43)} D a U ( t , x ) ≤ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( U ( t , x )) ) for 0 < a ≤ 1 , (43) and
D a U ( t , x ) ≥ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( U ( t , x ) ) ) for a ≥ 1. (44) D_aU(t,x)\geq\phi_*\left(\lambda_*^{-1}\ln a+\phi_*^{-1}(U(t,x))\right)\qquad\text{for }a\geq1. \tag*{(44)} D a U ( t , x ) ≥ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( U ( t , x )) ) for a ≥ 1. (44) Moreover,
D a U ( t , x ) = a U ( t , x ) + o ( U ( t , x ) ) as U ( t , x ) ↘ 0 , (45) D_aU(t,x)=aU(t,x)+o(U(t,x))\qquad\text{as }U(t,x)\searrow0, \tag*{(45)} D a U ( t , x ) = a U ( t , x ) + o ( U ( t , x )) as U ( t , x ) ↘ 0 , (45) uniformly with respect to the entire solution 0 < U < 1 0<U<1 0 < U < 1 ( t , x ) ∈ R × R N (t,x)\in\mathbb{R}\times\mathbb{R}^{N} ( t , x ) ∈ R × R N a ∈ ( 0 , ∞ ) a\in(0,\infty) a ∈ ( 0 , ∞ )
Proof. Fix ( t , x ) (t,x) ( t , x ) G ( a ) = D a U ( t , x ) G(a)=D_aU(t,x) G ( a ) = D a U ( t , x ) u ( s ) = G ( e s ) u(s)=G(e^s) u ( s ) = G ( e s ) w ( s ) = e s G + ′ ( e s ) w(s)=e^sG_+'(e^s) w ( s ) = e s G + ′ ( e s ) G + ′ G_+' G + ′ G G G G G G u u u w w w u ′ = w u'=w u ′ = w a = e s a=e^s a = e s (42) , with γ = 1 \gamma=1 γ = 1
D w ≤ ( 2 w − f ( u ) f ′ ( 0 ) ) d s Dw\leq\left(2w-\frac{f(u)}{f'(0)}\right)\,ds D w ≤ ( 2 w − f ′ ( 0 ) f ( u ) ) d s in the sense of distributions. The endpoints are u ( − ∞ ) = 0 u(-\infty)=0 u ( − ∞ ) = 0 u ( + ∞ ) = 1 u(+\infty)=1 u ( + ∞ ) = 1 q ( u ) q(u) q ( u ) d = w − q ( u ) / λ ∗ d=w-q(u)/\lambda_* d = w − q ( u ) / λ ∗
D d ≤ f ( u ) λ ∗ q ( u ) d d s . Dd\leq\frac{f(u)}{\lambda_*q(u)}d\,ds. D d ≤ λ ∗ q ( u ) f ( u ) d d s . Since d d d d ( s 0 ) = − δ < 0 d(s_0)=-\delta<0 d ( s 0 ) = − δ < 0
d ( s ) ≤ − δ exp ( ∫ s 0 s f ( u ( σ ) ) λ ∗ q ( u ( σ ) ) d σ ) ≤ − δ for s ≥ s 0 . d(s)\leq-\delta\exp\left(\int_{s_0}^{s}\frac{f(u(\sigma))}{\lambda_*q(u(\sigma))}\,d\sigma\right)\leq-\delta\qquad\text{for }s\geq s_0. d ( s ) ≤ − δ exp ( ∫ s 0 s λ ∗ q ( u ( σ )) f ( u ( σ )) d σ ) ≤ − δ for s ≥ s 0 . Consequently 0 ≤ w ( s ) ≤ q ( u ( s ) ) / λ ∗ − δ → − δ 0\leq w(s)\leq q(u(s))/\lambda_*-\delta\to-\delta 0 ≤ w ( s ) ≤ q ( u ( s )) / λ ∗ − δ → − δ s → + ∞ s\to+\infty s → + ∞ u ′ ≥ q ( u ) / λ ∗ u'\geq q(u)/\lambda_* u ′ ≥ q ( u ) / λ ∗
d d s ϕ ∗ − 1 ( u ( s ) ) = u ′ ( s ) q ( u ( s ) ) ≥ 1 λ ∗ for almost every s . \frac{d}{ds}\phi_*^{-1}(u(s))=\frac{u'(s)}{q(u(s))}\geq\frac{1}{\lambda_*}\qquad\text{for almost every }s. d s d ϕ ∗ − 1 ( u ( s )) = q ( u ( s )) u ′ ( s ) ≥ λ ∗ 1 for almost every s . For 0 < a ≤ 1 0<a\leq1 0 < a ≤ 1 [ ln a , 0 ] [\ln a,0] [ ln a , 0 ]
ϕ ∗ − 1 ( U ( t , x ) ) − ϕ ∗ − 1 ( D a U ( t , x ) ) ≥ − λ ∗ − 1 ln a . \phi_*^{-1}(U(t,x))-\phi_*^{-1}(D_aU(t,x))\geq-\lambda_*^{-1}\ln a. ϕ ∗ − 1 ( U ( t , x )) − ϕ ∗ − 1 ( D a U ( t , x )) ≥ − λ ∗ − 1 ln a . For a ≥ 1 a\geq1 a ≥ 1 [ 0 , ln a ] [0,\ln a] [ 0 , ln a ]
ϕ ∗ − 1 ( D a U ( t , x ) ) − ϕ ∗ − 1 ( U ( t , x ) ) ≥ λ ∗ − 1 ln a . \phi_*^{-1}(D_aU(t,x))-\phi_*^{-1}(U(t,x))\geq\lambda_*^{-1}\ln a. ϕ ∗ − 1 ( D a U ( t , x )) − ϕ ∗ − 1 ( U ( t , x )) ≥ λ ∗ − 1 ln a . Applying the increasing function ϕ ∗ \phi_* ϕ ∗ (43) and (44) . Finally
ϕ ∗ ( z + λ ∗ − 1 ln a ) a ϕ ∗ ( z ) ⟶ 1 ( z → − ∞ ) \frac{\phi_*(z+\lambda_*^{-1}\ln a)}{a\phi_*(z)}\longrightarrow1\qquad(z\to-\infty) a ϕ ∗ ( z ) ϕ ∗ ( z + λ ∗ − 1 ln a ) ⟶ 1 ( z → − ∞ ) uniformly for a a a (41) proves (45) .
The preceding amplitude estimates imply that superposition agrees with ordinary addition to first order when the solutions are small.
Lemma 2.7. Uniformly over all U , V ∈ C U,V\in\mathcal{C} U , V ∈ C
U ⊕ V = U + V + o ( U + V ) as U + V ↘ 0. U\oplus V=U+V+o(U+V)\qquad\text{as }U+V\searrow0. U ⊕ V = U + V + o ( U + V ) as U + V ↘ 0. Proof. Concavity of each Cauchy operator gives
1 2 ( D 2 U + D 2 V ) ≤ U ⊕ V ≤ U + V . \frac{1}{2}(D_2U+D_2V)\leq U\oplus V\leq U+V. 2 1 ( D 2 U + D 2 V ) ≤ U ⊕ V ≤ U + V . Applying (45) gives the conclusion.
Asymptotic uniqueness and algebraic laws
We first prove that suitable comparisons in the remote past determine the order of entire solutions. We use this result to establish the algebraic laws of dilation and superposition, construct subtraction, and prove the amplitude regularity needed below.
Write
F 1 ( u ) = f ′ ( 0 ) u − f ( u ) , F 2 ( u ) = f ( u ) − u f ′ ( u ) , (46) F_1(u)=f'(0)u-f(u),\qquad F_2(u)=f(u)-uf'(u), \tag*{(46)} F 1 ( u ) = f ′ ( 0 ) u − f ( u ) , F 2 ( u ) = f ( u ) − u f ′ ( u ) , (46) both nonnegative on [ 0 , 1 ] [0,1] [ 0 , 1 ] U U U (1) with 0 < U < 1 0<U<1 0 < U < 1 P U = ∂ t − Δ − f ′ ( U ) P_U=\partial_t-\Delta-f'(U) P U = ∂ t − Δ − f ′ ( U ) P U ( t , s ) P_U(t,s) P U ( t , s ) P U v = 0 P_Uv=0 P U v = 0
( ∂ t − Δ − f ′ ( U ) ) U = F 2 ( U ) ≥ 0 , (\partial_t-\Delta-f'(U))U=F_2(U)\ge0, ( ∂ t − Δ − f ′ ( U )) U = F 2 ( U ) ≥ 0 , P U ( t , s ) [ U ( s ) ] ≤ U ( t ) . (47) P_U(t,s)[U(s)]\le U(t). \tag*{(47)} P U ( t , s ) [ U ( s )] ≤ U ( t ) . (47) To prove the comparison and uniqueness results below, we consider two entire solutions X , Y X,Y X , Y (1) with 0 < X , Y < 1 0<X,Y<1 0 < X , Y < 1 ( X − Y ) + = o ( Y ) (X-Y)_+=o(Y) ( X − Y ) + = o ( Y ) Y ↘ 0 Y\searrow0 Y ↘ 0 Y ω ( Y ) Y\omega(Y) Y ω ( Y ) ω ≥ 0 \omega\ge0 ω ≥ 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] ω ( 0 ) = 0 \omega(0)=0 ω ( 0 ) = 0 X ≤ Y X\le Y X ≤ Y
Lemma 2.8. For every entire solution U U U (1) with 0 < U < 1 0<U<1 0 < U < 1 ω : [ 0 , 1 ] → R \omega:[0,1]\to\mathbb{R} ω : [ 0 , 1 ] → R ω ( 0 ) = 0 \omega(0)=0 ω ( 0 ) = 0 ( t , x ) (t,x) ( t , x )
lim T → ∞ 1 T ∫ t − T t P U ( t , s ) [ U ( s ) ∣ ω ( U ( s ) ) ∣ ] ( x ) d s = 0. (48) \lim_{T\to\infty}\frac{1}{T}\int_{t-T}^{t}P_U(t,s)[U(s)|\omega(U(s))|](x)\,\mathrm{d}s=0. \tag*{(48)} T → ∞ lim T 1 ∫ t − T t P U ( t , s ) [ U ( s ) ∣ ω ( U ( s )) ∣ ] ( x ) d s = 0. (48) Proof. We first show that, for every fixed δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 )
lim T → ∞ 1 T ∫ t − T t P U ( t , s ) [ U ( s ) 1 { U ( s ) ≥ δ } ] ( x ) d s = 0. \lim_{T\to\infty}\frac{1}{T}\int_{t-T}^{t}P_U(t,s)[U(s)\mathbf{1}_{\{U(s)\ge\delta\}}](x)\,\mathrm{d}s=0. T → ∞ lim T 1 ∫ t − T t P U ( t , s ) [ U ( s ) 1 { U ( s ) ≥ δ } ] ( x ) d s = 0. To construct an auxiliary function controlling this integral, we need that F 1 / F 2 F_1/F_2 F 1 / F 2 [ c , 1 ] [c,1] [ c , 1 ] 0 < c < 1 0<c<1 0 < c < 1
( F 1 F 2 ) ( u ) : = { F 1 ( u ) / F 2 ( u ) , F 2 ( u ) > 0 , 0 , F 2 ( u ) = 0. (49) \left(\frac{F_1}{F_2}\right)(u):=
\begin{cases}
F_1(u)/F_2(u), & F_2(u)>0,\\
0, & F_2(u)=0.
\end{cases} \tag*{(49)} ( F 2 F 1 ) ( u ) := { F 1 ( u ) / F 2 ( u ) , 0 , F 2 ( u ) > 0 , F 2 ( u ) = 0. (49) This convention is consistent with F 1 ( u ) = 0 F_1(u)=0 F 1 ( u ) = 0 (20) , which read
F 1 ( u ) = ∫ 0 u ( u − v ) ( − f ′ ′ ( v ) ) d v , F 2 ( u ) = ∫ 0 u v ( − f ′ ′ ( v ) ) d v . F_1(u)=\int_0^u(u-v)(-f''(v))\,\mathrm{d}v,\qquad F_2(u)=\int_0^u v(-f''(v))\,\mathrm{d}v. F 1 ( u ) = ∫ 0 u ( u − v ) ( − f ′′ ( v )) d v , F 2 ( u ) = ∫ 0 u v ( − f ′′ ( v )) d v . If F 2 ( c ) > 0 F_2(c)>0 F 2 ( c ) > 0 F 2 ( u ) ≥ F 2 ( c ) F_2(u)\ge F_2(c) F 2 ( u ) ≥ F 2 ( c ) u ≥ c u\ge c u ≥ c θ ≥ c \theta\ge c θ ≥ c f ( u ) = f ′ ( 0 ) u f(u)=f'(0)u f ( u ) = f ′ ( 0 ) u θ < 1 \theta<1 θ < 1 f ( 1 ) = 0 < f ′ ( 0 ) f(1)=0<f'(0) f ( 1 ) = 0 < f ′ ( 0 ) F 1 F_1 F 1 F 2 F_2 F 2 [ 0 , θ ] [0,\theta] [ 0 , θ ] u > θ u>\theta u > θ
F 1 ( u ) ≤ ( u − θ ) ∫ θ u ( − f ′ ′ ( v ) ) d v , F 2 ( u ) ≥ θ ∫ θ u ( − f ′ ′ ( v ) ) d v > 0. F_1(u)\le(u-\theta)\int_\theta^u(-f''(v))\,\mathrm{d}v,\qquad F_2(u)\ge\theta\int_\theta^u(-f''(v))\,\mathrm{d}v>0. F 1 ( u ) ≤ ( u − θ ) ∫ θ u ( − f ′′ ( v )) d v , F 2 ( u ) ≥ θ ∫ θ u ( − f ′′ ( v )) d v > 0. Thus F 1 ( u ) / F 2 ( u ) ≤ ( u − θ ) / θ F_1(u)/F_2(u)\le(u-\theta)/\theta F 1 ( u ) / F 2 ( u ) ≤ ( u − θ ) / θ
Keep δ \delta δ d = ln δ d=\ln\delta d = ln δ ℓ = 1 − d \ell=1-d ℓ = 1 − d R > − d R>-d R > − d C 1 C^1 C 1 ( − ∞ , 0 ] (-\infty,0] ( − ∞ , 0 ]
Ψ R ′ ( h ) = { 0 , h ≤ − 2 R , ℓ ( h + 2 R ) / R , − 2 R < h < − R , ℓ , − R ≤ h < d , 1 − h , d ≤ h ≤ 0. \Psi_R'(h)=
\begin{cases}
0, & h\le-2R,\\
\ell(h+2R)/R, & -2R<h<-R,\\
\ell, & -R\le h<d,\\
1-h, & d\le h\le0.
\end{cases} Ψ R ′ ( h ) = ⎩ ⎨ ⎧ 0 , ℓ ( h + 2 R ) / R , ℓ , 1 − h , h ≤ − 2 R , − 2 R < h < − R , − R ≤ h < d , d ≤ h ≤ 0. Ψ R ( h ) = A R + ∫ − 2 R h Ψ R ′ ( z ) d z , A R = 1 + ℓ sup e − R ≤ u ≤ 1 F 2 ( u ) > 0 F 1 ( u ) F 2 ( u ) . \Psi_R(h)=A_R+\int_{-2R}^{h}\Psi'_R(z)\,\mathrm{d}z,\qquad A_R=1+\ell\sup_{\substack{e^{-R}\leq u\leq1\\F_2(u)>0}}\frac{F_1(u)}{F_2(u)}. Ψ R ( h ) = A R + ∫ − 2 R h Ψ R ′ ( z ) d z , A R = 1 + ℓ e − R ≤ u ≤ 1 F 2 ( u ) > 0 sup F 2 ( u ) F 1 ( u ) . The supremum defining A R A_R A R (49) on [ e − R , 1 ] [e^{-R},1] [ e − R , 1 ] Ψ R ≥ A R > 0 \Psi_R\geq A_R>0 Ψ R ≥ A R > 0 0 ≤ Ψ R ′ ≤ ℓ 0\leq\Psi'_R\leq\ell 0 ≤ Ψ R ′ ≤ ℓ A R A_R A R F 2 ( e h ) Ψ R ( h ) ≥ F 1 ( e h ) Ψ R ′ ( h ) F_2(e^h)\Psi_R(h)\geq F_1(e^h)\Psi'_R(h) F 2 ( e h ) Ψ R ( h ) ≥ F 1 ( e h ) Ψ R ′ ( h ) h ≥ − R h\geq-R h ≥ − R F 2 ( e h ) = 0 F_2(e^h)=0 F 2 ( e h ) = 0 F 1 ( e h ) = 0 F_1(e^h)=0 F 1 ( e h ) = 0 h = ln U h=\ln U h = ln U p = ∣ ∇ h ∣ 2 ≤ f ′ ( 0 ) p=|\nabla h|^2\leq f'(0) p = ∣∇ h ∣ 2 ≤ f ′ ( 0 ) Theorem 2.1 . For a function of h h h
A p Ψ = p Ψ ′ ′ + ( p − f ( U ) U ) Ψ ′ − F 2 ( U ) U Ψ = p Ψ ′ ′ + ( p − f ′ ( 0 ) ) Ψ ′ + F 1 ( U ) U Ψ ′ − F 2 ( U ) U Ψ . A_p\Psi=p\Psi''+\left(p-\frac{f(U)}{U}\right)\Psi'-\frac{F_2(U)}{U}\Psi=p\Psi''+\left(p-f'(0)\right)\Psi'+\frac{F_1(U)}{U}\Psi'-\frac{F_2(U)}{U}\Psi. A p Ψ = p Ψ ′′ + ( p − U f ( U ) ) Ψ ′ − U F 2 ( U ) Ψ = p Ψ ′′ + ( p − f ′ ( 0 ) ) Ψ ′ + U F 1 ( U ) Ψ ′ − U F 2 ( U ) Ψ. On h ≥ d h\geq d h ≥ d − f ′ ( 0 ) + ( f ′ ( 0 ) − p ) h ≤ − f ′ ( 0 ) -f'(0)+(f'(0)-p)h\leq-f'(0) − f ′ ( 0 ) + ( f ′ ( 0 ) − p ) h ≤ − f ′ ( 0 ) − R ≤ h < d -R\leq h<d − R ≤ h < d Ψ R ′ ′ = 0 \Psi''_R=0 Ψ R ′′ = 0 ( p − f ′ ( 0 ) ) Ψ R ′ ≤ 0 (p-f'(0))\Psi'_R\leq0 ( p − f ′ ( 0 )) Ψ R ′ ≤ 0 − 2 R < h < − R -2R<h<-R − 2 R < h < − R Ψ R ′ ′ = ℓ / R \Psi''_R=\ell/R Ψ R ′′ = ℓ / R 0 ≤ Ψ R ′ ≤ ℓ 0\leq\Psi'_R\leq\ell 0 ≤ Ψ R ′ ≤ ℓ F 2 Ψ R ≥ 0 F_2\Psi_R\geq0 F 2 Ψ R ≥ 0
ε R = f ′ ( 0 ) ℓ / R + ℓ sup 0 < u ≤ e − R F 1 ( u ) / u → 0. \varepsilon_R=f'(0)\ell/R+\ell\sup_{0<u\leq e^{-R}}F_1(u)/u\to0. ε R = f ′ ( 0 ) ℓ / R + ℓ 0 < u ≤ e − R sup F 1 ( u ) / u → 0. Here ε R → 0 \varepsilon_R\to0 ε R → 0 R → ∞ R\to\infty R → ∞ δ \delta δ F 1 ( u ) / u → 0 F_1(u)/u\to0 F 1 ( u ) / u → 0 u ↘ 0 u\searrow0 u ↘ 0 h < − 2 R h<-2R h < − 2 R Ψ R \Psi_R Ψ R A p Ψ R = − F 2 ( U ) Ψ R / U ≤ 0 A_p\Psi_R=-F_2(U)\Psi_R/U\leq0 A p Ψ R = − F 2 ( U ) Ψ R / U ≤ 0
A p Ψ R ≤ − f ′ ( 0 ) 1 { U ≥ δ } + ε R . A_p\Psi_R\leq-f'(0)1_{\{U\geq\delta\}}+\varepsilon_R. A p Ψ R ≤ − f ′ ( 0 ) 1 { U ≥ δ } + ε R . For F R = U Ψ R ( ln U ) F_R=U\Psi_R(\ln U) F R = U Ψ R ( ln U )
( ∂ t − Δ − f ′ ( U ) ) F R = − U A p Ψ R ≥ f ′ ( 0 ) U 1 { U ≥ δ } − ε R U . (\partial_t-\Delta-f'(U))F_R=-UA_p\Psi_R\geq f'(0)U1_{\{U\geq\delta\}}-\varepsilon_RU. ( ∂ t − Δ − f ′ ( U )) F R = − U A p Ψ R ≥ f ′ ( 0 ) U 1 { U ≥ δ } − ε R U . The differential inequality holds in the sense of distributions. Indeed, u ↦ u Ψ R ( ln u ) u\mapsto u\Psi_R(\ln u) u ↦ u Ψ R ( ln u ) [ 0 , 1 ] [0,1] [ 0 , 1 ] A R u A_Ru A R u u ≤ e − 2 R u\leq e^{-2R} u ≤ e − 2 R P U P_U P U
f ′ ( 0 ) ∫ t − T t P U ( t , s ) [ U ( s ) 1 { U ( s ) ≥ δ } ] ( x ) d s ≤ F R ( t , x ) − ( P U ( t , t − T ) [ F R ( t − T ) ] ) ( x ) + ε R ∫ t − T t P U ( t , s ) [ U ( s ) ] ( x ) d s ≤ F R ( t , x ) + ε R T U ( t , x ) . \begin{aligned}
f'(0)\int_{t-T}^{t}P_U(t,s)[U(s)1_{\{U(s)\geq\delta\}}](x)\,\mathrm{d}s\leq F_R(t,x)-(P_U(t,t-T)[F_R(t-T)])(x)+\varepsilon_R\int_{t-T}^{t}P_U(t,s)[U(s)](x)\,\mathrm{d}s \\
&\leq F_R(t,x)+\varepsilon_RTU(t,x).
\end{aligned} f ′ ( 0 ) ∫ t − T t P U ( t , s ) [ U ( s ) 1 { U ( s ) ≥ δ } ] ( x ) d s ≤ F R ( t , x ) − ( P U ( t , t − T ) [ F R ( t − T )]) ( x ) + ε R ∫ t − T t P U ( t , s ) [ U ( s )] ( x ) d s ≤ F R ( t , x ) + ε R T U ( t , x ) . The last inequality uses F R ≥ 0 F_R\geq0 F R ≥ 0 (47) . Dividing by f ′ ( 0 ) T f'(0)T f ′ ( 0 ) T
1 T ∫ t − T t P U ( t , s ) [ U ( s ) 1 { U ( s ) ≥ δ } ] ( x ) d s ≤ F R ( t , x ) f ′ ( 0 ) T + ε R f ′ ( 0 ) U ( t , x ) . \frac{1}{T}\int_{t-T}^{t}P_U(t,s)[U(s)1_{\{U(s)\geq\delta\}}](x)\,\mathrm{d}s\leq\frac{F_R(t,x)}{f'(0)T}+\frac{\varepsilon_R}{f'(0)}U(t,x). T 1 ∫ t − T t P U ( t , s ) [ U ( s ) 1 { U ( s ) ≥ δ } ] ( x ) d s ≤ f ′ ( 0 ) T F R ( t , x ) + f ′ ( 0 ) ε R U ( t , x ) . For fixed R R R F R ( t , x ) F_R(t,x) F R ( t , x ) T → ∞ T\to\infty T → ∞ R → ∞ R\to\infty R → ∞ { U ≥ δ } \{U\geq\delta\} { U ≥ δ }
Finally, split U ∣ ω ( U ) ∣ U|\omega(U)| U ∣ ω ( U ) ∣ { U < δ } \{U<\delta\} { U < δ } { U ≥ δ } \{U\geq\delta\} { U ≥ δ } (47) , the averaged contribution of the first set is at most U ( t , x ) sup [ 0 , δ ] ∣ ω ∣ U(t,x)\sup_{[0,\delta]}|\omega| U ( t , x ) sup [ 0 , δ ] ∣ ω ∣ ∥ ω ∥ L ∞ ( [ 0 , 1 ] ) \|\omega\|_{L^\infty([0,1])} ∥ ω ∥ L ∞ ([ 0 , 1 ])
lim sup T → ∞ 1 T ∫ t − T t P U ( t , s ) [ U ( s ) ∣ ω ( U ( s ) ) ∣ ] ( x ) d s ≤ U ( t , x ) sup [ 0 , δ ] ∣ ω ∣ . \limsup_{T\to\infty}\frac{1}{T}\int_{t-T}^{t}P_U(t,s)[U(s)|\omega(U(s))|](x)\,\mathrm{d}s\leq U(t,x)\sup_{[0,\delta]}|\omega|. T → ∞ lim sup T 1 ∫ t − T t P U ( t , s ) [ U ( s ) ∣ ω ( U ( s )) ∣ ] ( x ) d s ≤ U ( t , x ) [ 0 , δ ] sup ∣ ω ∣. Letting δ ↘ 0 \delta\searrow0 δ ↘ 0 ω \omega ω ω ( 0 ) = 0 \omega(0)=0 ω ( 0 ) = 0
The following theorem shows that a sufficiently small error relative to an entire solution near zero determines the order of the solutions and, in particular, yields uniqueness.
Theorem 2.9. If entire solutions X , Y X,Y X , Y (1) with 0 < X , Y < 1 0<X,Y<1 0 < X , Y < 1 ( X − Y ) + ≤ Y ω ( Y ) (X-Y)_{+}\leq Y\omega(Y) ( X − Y ) + ≤ Y ω ( Y ) ω ≥ 0 \omega\geq0 ω ≥ 0 ω ( 0 ) = 0 \omega(0)=0 ω ( 0 ) = 0 X ≤ Y X\leq Y X ≤ Y Z Z Z c Z ≤ X , Y ≤ C Z cZ\leq X,Y\leq CZ c Z ≤ X , Y ≤ C Z c , C > 0 c,C>0 c , C > 0 ∣ X − Y ∣ = o ( Z ) \lvert X-Y\rvert=o(Z) ∣ X − Y ∣ = o ( Z ) Z ↘ 0 Z\searrow0 Z ↘ 0 X = Y X=Y X = Y
Proof. Concavity and Kato’s inequality give
( ∂ t − Δ ) ( X − Y ) + ≤ f ′ ( Y ) ( X − Y ) + . (\partial_t-\Delta)(X-Y)_{+}\leq f'(Y)(X-Y)_{+}. ( ∂ t − Δ ) ( X − Y ) + ≤ f ′ ( Y ) ( X − Y ) + . Hence ( X − Y ) + ( t ) ≤ P Y ( t , s ) [ Y ( s ) ω ( Y ( s ) ) ] (X-Y)_{+}(t)\leq P_Y(t,s)[Y(s)\omega(Y(s))] ( X − Y ) + ( t ) ≤ P Y ( t , s ) [ Y ( s ) ω ( Y ( s ))] s < t s<t s < t s ∈ [ t − T , t ] s\in[t-T,t] s ∈ [ t − T , t ] Lemma 2.8 . For the second assertion, Z ≤ Y / c Z\leq Y/c Z ≤ Y / c Z Z Z
∣ X − Y ∣ Y ≤ 1 c ∣ X − Y ∣ Z → 0 uniformly as Y ↘ 0. \frac{\lvert X-Y\rvert}{Y}\leq\frac{1}{c}\frac{\lvert X-Y\rvert}{Z}\to0\qquad\text{uniformly as }Y\searrow0. Y ∣ X − Y ∣ ≤ c 1 Z ∣ X − Y ∣ → 0 uniformly as Y ↘ 0. Moreover, X ≤ C Z ≤ ( C / c ) Y X\leq CZ\leq(C/c)Y X ≤ C Z ≤ ( C / c ) Y ∣ X − Y ∣ / Y ≤ C / c + 1 \lvert X-Y\rvert/Y\leq C/c+1 ∣ X − Y ∣ / Y ≤ C / c + 1 ω ≥ 0 \omega\geq0 ω ≥ 0 ω ( 0 ) = 0 \omega(0)=0 ω ( 0 ) = 0 ∣ X − Y ∣ ≤ Y ω ( Y ) \lvert X-Y\rvert\leq Y\omega(Y) ∣ X − Y ∣ ≤ Y ω ( Y ) X ≤ Y X\leq Y X ≤ Y X X X Y Y Y Y ≤ X Y\leq X Y ≤ X X = Y X=Y X = Y
Theorem 2.9 allows us to establish the algebraic laws of nonlinear dilation and superposition, analogous to those of scalar multiplication and addition.
Proposition 2.10. For a , b > 0 a,b>0 a , b > 0 U , V , W ∈ C U,V,W\in\mathcal{C} U , V , W ∈ C
D a D b U = D a b U , D 1 U = U , ( U ⊕ V ) ⊕ W = U ⊕ ( V ⊕ W ) , U ⊕ V = V ⊕ U , D a ( U ⊕ V ) = D a U ⊕ D a V , D a U ⊕ D b U = D a + b U . (50) \begin{aligned}
D_aD_bU&=D_{ab}U, & D_1U&=U, \\
(U\oplus V)\oplus W&=U\oplus(V\oplus W), & U\oplus V&=V\oplus U, \\
D_a(U\oplus V)&=D_aU\oplus D_aV, & D_aU\oplus D_bU&=D_{a+b}U. \tag*{(50)}
\end{aligned} D a D b U ( U ⊕ V ) ⊕ W D a ( U ⊕ V ) = D ab U , = U ⊕ ( V ⊕ W ) , = D a U ⊕ D a V , D 1 U U ⊕ V D a U ⊕ D b U = U , = V ⊕ U , = D a + b U . (50) Thus D 1 / a D_{1/a} D 1/ a D a D_a D a
Proof. We apply the second assertion of Theorem 2.9 to each pair of entire solutions below. The bounds (41) make both members of each pair comparable to the indicated positive function, with constants depending only on a a a b b b
For the dilation group law, take Z = U Z=U Z = U (45) ,
D a ( D b U ) = a D b U + o ( D b U ) = a b U + o ( U ) , D a b U = a b U + o ( U ) as U ↘ 0. D_a(D_bU)=aD_bU+o(D_bU)=abU+o(U),\qquad D_{ab}U=abU+o(U)\quad\text{as }U\searrow0. D a ( D b U ) = a D b U + o ( D b U ) = ab U + o ( U ) , D ab U = ab U + o ( U ) as U ↘ 0. Theorem 2.9 therefore gives D a D b U = D a b U D_aD_bU=D_{ab}U D a D b U = D ab U D 1 U = U D_1U=U D 1 U = U U U U
For associativity, take Z = U + V + W Z=U+V+W Z = U + V + W max ( U , V ) ≥ ( U + V ) / 2 \max(U,V)\geq(U+V)/2 max ( U , V ) ≥ ( U + V ) /2
1 4 Z ≤ ( U ⊕ V ) ⊕ W , U ⊕ ( V ⊕ W ) ≤ Z . \frac{1}{4}Z\leq(U\oplus V)\oplus W,\ U\oplus(V\oplus W)\leq Z. 4 1 Z ≤ ( U ⊕ V ) ⊕ W , U ⊕ ( V ⊕ W ) ≤ Z . Applying Lemma 2.7 twice yields
( U ⊕ V ) ⊕ W = Z + o ( Z ) , U ⊕ ( V ⊕ W ) = Z + o ( Z ) as Z ↘ 0. (U\oplus V)\oplus W=Z+o(Z),\qquad U\oplus(V\oplus W)=Z+o(Z)\quad\text{as }Z\searrow0. ( U ⊕ V ) ⊕ W = Z + o ( Z ) , U ⊕ ( V ⊕ W ) = Z + o ( Z ) as Z ↘ 0. The same theorem proves associativity. Commutativity follows directly from the definition (40) .
For distributivity, take Z = U + V Z=U+V Z = U + V (45) with Lemma 2.7 , we obtain
D a ( U ⊕ V ) = a Z + o ( Z ) , D a U ⊕ D a V = a Z + o ( Z ) as Z ↘ 0. D_a(U\oplus V)=aZ+o(Z),\qquad D_aU\oplus D_aV=aZ+o(Z)\quad\text{as }Z\searrow0. D a ( U ⊕ V ) = a Z + o ( Z ) , D a U ⊕ D a V = a Z + o ( Z ) as Z ↘ 0. Theorem 2.9 gives D a ( U ⊕ V ) = D a U ⊕ D a V D_a(U \oplus V)=D_aU \oplus D_aV D a ( U ⊕ V ) = D a U ⊕ D a V Z = U Z=U Z = U
D a U ⊕ D b U = ( a + b ) U + o ( U ) , D a + b U = ( a + b ) U + o ( U ) as U ↘ 0. D_aU \oplus D_bU=(a+b)U+o(U), \qquad D_{a+b}U=(a+b)U+o(U) \quad\text{as } U\searrow0. D a U ⊕ D b U = ( a + b ) U + o ( U ) , D a + b U = ( a + b ) U + o ( U ) as U ↘ 0. Another application of Theorem 2.9 proves the last identity. If any of U , V , W U,V,W U , V , W D a 0 = 0 D_a0=0 D a 0 = 0 U ⊕ 0 = U U\oplus0=U U ⊕ 0 = U D 1 / a D a = D a D 1 / a = D 1 D_{1/a}D_a=D_aD_{1/a}=D_1 D 1/ a D a = D a D 1/ a = D 1 D a D_a D a
We next prove an order cancellation law: adding the same solution by nonlinear superposition preserves and reflects the order.
Proposition 2.11. For U , V , W ∈ C U,V,W\in\mathcal{C} U , V , W ∈ C
U ⊕ W ≤ V ⊕ W ⟺ U ≤ V . U\oplus W\le V\oplus W \quad\Longleftrightarrow\quad U\le V. U ⊕ W ≤ V ⊕ W ⟺ U ≤ V . In particular, U ⊕ W = V ⊕ W U\oplus W=V\oplus W U ⊕ W = V ⊕ W U = V U=V U = V
Proof. The implication from right to left follows from order preservation. Conversely, assume U ⊕ W ≤ V ⊕ W U\oplus W\le V\oplus W U ⊕ W ≤ V ⊕ W n ≥ 1 n\ge1 n ≥ 1 0 ≤ j < n 0\le j<n 0 ≤ j < n
U ⊕ ⋯ ⊕ U ⏟ n − j ⊕ V ⊕ ⋯ ⊕ V ⏟ j ⊕ W ≤ U ⊕ ⋯ ⊕ U ⏟ n − j − 1 ⊕ V ⊕ ⋯ ⊕ V ⏟ j + 1 ⊕ W . \underbrace{U\oplus\cdots\oplus U}_{n-j}\oplus\underbrace{V\oplus\cdots\oplus V}_{j}\oplus W
\le
\underbrace{U\oplus\cdots\oplus U}_{n-j-1}\oplus\underbrace{V\oplus\cdots\oplus V}_{j+1}\oplus W. n − j U ⊕ ⋯ ⊕ U ⊕ j V ⊕ ⋯ ⊕ V ⊕ W ≤ n − j − 1 U ⊕ ⋯ ⊕ U ⊕ j + 1 V ⊕ ⋯ ⊕ V ⊕ W . Iterating over j j j
U ⊕ ⋯ ⊕ U ⏟ n ⊕ W ≤ V ⊕ ⋯ ⊕ V ⏟ n ⊕ W . \underbrace{U\oplus\cdots\oplus U}_{n}\oplus W
\le
\underbrace{V\oplus\cdots\oplus V}_{n}\oplus W. n U ⊕ ⋯ ⊕ U ⊕ W ≤ n V ⊕ ⋯ ⊕ V ⊕ W . Applying the order-preserving map D 1 / n D_{1/n} D 1/ n
U ⊕ D 1 / n W ≤ V ⊕ D 1 / n W . U\oplus D_{1/n}W\le V\oplus D_{1/n}W. U ⊕ D 1/ n W ≤ V ⊕ D 1/ n W . The elementary bounds for superposition imply
U ≤ U ⊕ D 1 / n W ≤ V ⊕ D 1 / n W ≤ V + D 1 / n W . U\le U\oplus D_{1/n}W\le V\oplus D_{1/n}W\le V+D_{1/n}W. U ≤ U ⊕ D 1/ n W ≤ V ⊕ D 1/ n W ≤ V + D 1/ n W . Letting n → ∞ n\to\infty n → ∞ U ≤ V U\le V U ≤ V
The dilation group law also makes the amplitude parameter a a a
Corollary 2.12. For an entire solution U ∈ ( 0 , 1 ) U\in(0,1) U ∈ ( 0 , 1 ) ( t , x ) (t,x) ( t , x ) a ↦ D a U ( t , x ) a\mapsto D_aU(t,x) a ↦ D a U ( t , x ) ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) ( 0 , 1 ) (0,1) ( 0 , 1 ) 0 < a < b 0<a<b 0 < a < b
0 ≤ D b U − D a U ≤ ( b / a − 1 ) D a U ≤ ( b − a ) / a . (51) 0\le D_bU-D_aU\le(b/a-1)D_aU\le(b-a)/a. \tag*{(51)} 0 ≤ D b U − D a U ≤ ( b / a − 1 ) D a U ≤ ( b − a ) / a . (51) Proof. Concavity and order preservation give continuity and monotonicity in a a a 0 < a < b 0<a<b 0 < a < b D a U ( t 0 , x 0 ) = D b U ( t 0 , x 0 ) D_aU(t_0,x_0)=D_bU(t_0,x_0) D a U ( t 0 , x 0 ) = D b U ( t 0 , x 0 ) D a U ≡ D b U D_aU\equiv D_bU D a U ≡ D b U Lemma 2.4 ,
U = D a / b U = D ( a / b ) n U → 0 for n → ∞ , U=D_{a/b}U=D_{(a/b)^n}U\to0 \quad\text{for } n\to\infty, U = D a / b U = D ( a / b ) n U → 0 for n → ∞ , contradicting U > 0 U>0 U > 0 D a U ( t , x ) < D b U ( t , x ) D_aU(t,x)<D_bU(t,x) D a U ( t , x ) < D b U ( t , x ) ( t , x ) (t,x) ( t , x )
Lemma 2.4 gives
lim a ↘ 0 D a U ( t , x ) = 0 , lim a → ∞ D a U ( t , x ) = 1. \lim_{a\searrow0}D_aU(t,x)=0,\qquad\lim_{a\to\infty}D_aU(t,x)=1. a ↘ 0 lim D a U ( t , x ) = 0 , a → ∞ lim D a U ( t , x ) = 1. Together with continuity and strict monotonicity, these limits prove the assertion. The Lipschitz estimate follows from the group law and D b / a Z ≤ ( b / a ) Z D_{b/a}Z\le(b/a)Z D b / a Z ≤ ( b / a ) Z
We now show that nonlinear dilation commutes with monotone limits of entire solutions.
Proposition 2.13. Suppose U n ∈ C U_n \in\mathcal{C} U n ∈ C U ∈ C U \in\mathcal{C} U ∈ C a > 0 a > 0 a > 0
D a U n → D a U in C l o c 1 , 2 ( R × R N ) as n → ∞ . D_aU_n \to D_aU \quad\text{in } C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N) \quad\text{as } n \to\infty. D a U n → D a U in C loc 1 , 2 ( R × R N ) as n → ∞. Proof. Fix a > 0 a > 0 a > 0 0 ≤ D a U n ≤ 1 0 \leq D_aU_n \leq1 0 ≤ D a U n ≤ 1 V = lim n → ∞ D a U n V = \lim_{n \to\infty} D_aU_n V = lim n → ∞ D a U n V ∈ E V \in\mathcal{E} V ∈ E
D a U n → V in C l o c 1 , 2 ( R × R N ) . D_aU_n \to V \quad\text{in } C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N). D a U n → V in C loc 1 , 2 ( R × R N ) . For a decreasing sequence V ≤ D a U 1 ∈ C V \leq D_aU_1 \in\mathcal{C} V ≤ D a U 1 ∈ C V ≤ D a U ∈ C V \leq D_aU \in\mathcal{C} V ≤ D a U ∈ C V ∈ C V \in\mathcal{C} V ∈ C V V V U n ↘ U U_n \searrow U U n ↘ U
D a U ≤ V ≤ D a U n . D_aU \leq V \leq D_aU_n. D a U ≤ V ≤ D a U n . Applying the order-preserving map D 1 / a D_{1/a} D 1/ a
U ≤ D 1 / a V ≤ U n . U \leq D_{1/a}V \leq U_n. U ≤ D 1/ a V ≤ U n . Letting n → ∞ n \to\infty n → ∞ D 1 / a V = U D_{1/a}V = U D 1/ a V = U U n ↗ U U_n \nearrow U U n ↗ U
D a U n ≤ V ≤ D a U ⇒ U n ≤ D 1 / a V ≤ U , D_aU_n \leq V \leq D_aU \quad\Rightarrow\quad U_n \leq D_{1/a}V \leq U, D a U n ≤ V ≤ D a U ⇒ U n ≤ D 1/ a V ≤ U , and again D 1 / a V = U D_{1/a}V = U D 1/ a V = U n → ∞ n \to\infty n → ∞ V = D a ( D 1 / a V ) = D a U V = D_a(D_{1/a}V) = D_aU V = D a ( D 1/ a V ) = D a U a > 0 a > 0 a > 0
We next extend monotone continuity from dilation to superposition for decreasing sequences.
Lemma 2.14. If U n ↘ U U_n \searrow U U n ↘ U V n ↘ V V_n \searrow V V n ↘ V C \mathcal{C} C
U n ⊕ V n ↘ U ⊕ V in C l o c 1 , 2 ( R × R N ) as n → ∞ . U_n \oplus V_n \searrow U \oplus V \quad\text{in } C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N) \quad\text{as } n \to\infty. U n ⊕ V n ↘ U ⊕ V in C loc 1 , 2 ( R × R N ) as n → ∞. Proof. For fixed ( t , x ) (t,x) ( t , x ) ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ ) ] ) ( x ) (S_{t+T}[U_n(-T,\cdot) + V_n(-T,\cdot)])(x) ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ )]) ( x ) n n n T > max { 0 , − t } T > \max\{0,-t\} T > max { 0 , − t }
lim n → ∞ ( U n ⊕ V n ) ( t , x ) = lim n → ∞ lim T → ∞ ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ ) ] ) ( x ) = lim T → ∞ lim n → ∞ ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ ) ] ) ( x ) = lim T → ∞ ( S t + T [ U ( − T , ⋅ ) + V ( − T , ⋅ ) ] ) ( x ) = ( U ⊕ V ) ( t , x ) . \begin{aligned}
\lim_{n \to\infty}(U_n \oplus V_n)(t,x)
&= \lim_{n \to\infty}\lim_{T \to\infty}(S_{t+T}[U_n(-T,\cdot) + V_n(-T,\cdot)])(x) \\
&= \lim_{T \to\infty}\lim_{n \to\infty}(S_{t+T}[U_n(-T,\cdot) + V_n(-T,\cdot)])(x) \\
&= \lim_{T \to\infty}(S_{t+T}[U(-T,\cdot) + V(-T,\cdot)])(x) \\
&= (U \oplus V)(t,x).
\end{aligned} n → ∞ lim ( U n ⊕ V n ) ( t , x ) = n → ∞ lim T → ∞ lim ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ )]) ( x ) = T → ∞ lim n → ∞ lim ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ )]) ( x ) = T → ∞ lim ( S t + T [ U ( − T , ⋅ ) + V ( − T , ⋅ )]) ( x ) = ( U ⊕ V ) ( t , x ) . Parabolic compactness gives convergence in C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N) C loc 1 , 2 ( R × R N )
For general locally convergent sequences, we record the one-sided limit inequalities needed in the construction of subtraction.
Lemma 2.15. Suppose U n , V n ∈ C U_n,V_n \in\mathcal{C} U n , V n ∈ C U , V ∈ C U,V \in\mathcal{C} U , V ∈ C ( t , x ) (t,x) ( t , x )
D a U ≤ lim inf n D a U n for 0 < a < 1 , D a U ≥ lim sup n D a U n for a > 1 , D_aU \leq\liminf_{n} D_aU_n \quad\text{for } 0 < a < 1, \qquad D_aU \geq\limsup_{n} D_aU_n \quad\text{for } a > 1, D a U ≤ n lim inf D a U n for 0 < a < 1 , D a U ≥ n lim sup D a U n for a > 1 , U ⊕ V ≥ lim sup n ( U n ⊕ V n ) . U \oplus V \geq\limsup_{n}(U_n \oplus V_n). U ⊕ V ≥ n lim sup ( U n ⊕ V n ) . Proof. Fix ( t , x ) (t,x) ( t , x ) T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t }
lim inf n → ∞ ( D a U n ) ( t , x ) ≥ lim n → ∞ ( S t + T [ a U n ( − T , ⋅ ) ] ) ( x ) = ( S t + T [ a U ( − T , ⋅ ) ] ) ( x ) for 0 < a < 1 , lim sup n → ∞ ( D a U n ) ( t , x ) ≤ lim n → ∞ ( S t + T [ a U n ( − T , ⋅ ) ] ) ( x ) = ( S t + T [ a U ( − T , ⋅ ) ] ) ( x ) for a > 1 , lim sup n → ∞ ( U n ⊕ V n ) ( t , x ) ≤ lim n → ∞ ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ ) ] ) ( x ) = ( S t + T [ U ( − T , ⋅ ) + V ( − T , ⋅ ) ] ) ( x ) . \begin{aligned}
\liminf_{n\to\infty}(D_aU_n)(t,x)&\ge\lim_{n\to\infty}(S_{t+T}[aU_n(-T,\cdot)])(x)\\
&=(S_{t+T}[aU(-T,\cdot)])(x) &&\text{for }0<a<1,\\
\limsup_{n\to\infty}(D_aU_n)(t,x)&\le\lim_{n\to\infty}(S_{t+T}[aU_n(-T,\cdot)])(x)\\
&=(S_{t+T}[aU(-T,\cdot)])(x) &&\text{for }a>1,\\
\limsup_{n\to\infty}(U_n\oplus V_n)(t,x)&\le\lim_{n\to\infty}(S_{t+T}[U_n(-T,\cdot)+V_n(-T,\cdot)])(x)\\
&=(S_{t+T}[U(-T,\cdot)+V(-T,\cdot)])(x).
\end{aligned} n → ∞ lim inf ( D a U n ) ( t , x ) n → ∞ lim sup ( D a U n ) ( t , x ) n → ∞ lim sup ( U n ⊕ V n ) ( t , x ) ≥ n → ∞ lim ( S t + T [ a U n ( − T , ⋅ )]) ( x ) = ( S t + T [ a U ( − T , ⋅ )]) ( x ) ≤ n → ∞ lim ( S t + T [ a U n ( − T , ⋅ )]) ( x ) = ( S t + T [ a U ( − T , ⋅ )]) ( x ) ≤ n → ∞ lim ( S t + T [ U n ( − T , ⋅ ) + V n ( − T , ⋅ )]) ( x ) = ( S t + T [ U ( − T , ⋅ ) + V ( − T , ⋅ )]) ( x ) . for 0 < a < 1 , for a > 1 , Letting T → ∞ T\to\infty T → ∞
The cancellation law gives uniqueness of a nonlinear difference. The following theorem establishes its existence whenever the two solutions are ordered.
Theorem 2.16. If U , V ∈ C U,V\in\mathcal{C} U , V ∈ C U ≤ V U\le V U ≤ V W ∈ C W\in\mathcal{C} W ∈ C U ⊕ W = V U\oplus W=V U ⊕ W = V increasing pullback from V − U V-U V − U
Proof. The cases U = 0 U=0 U = 0 U = V U=V U = V 0 < U < V 0<U<V 0 < U < V b > 0 b>0 b > 0 F b = ( 1 + b ) V − b U F_b=(1+b)V-bU F b = ( 1 + b ) V − b U V = ( F b + b U ) / ( 1 + b ) V=(F_b+bU)/(1+b) V = ( F b + b U ) / ( 1 + b ) A b A_b A b
V ≤ A b ≤ F b ≤ ( 1 + b ) V , A b + D b U ≥ D 1 + b V . V\le A_b\le F_b\le(1+b)V,\qquad A_b+D_bU\ge D_{1+b}V. V ≤ A b ≤ F b ≤ ( 1 + b ) V , A b + D b U ≥ D 1 + b V . The last inequality follows from subadditivity since F b + b U = ( 1 + b ) V F_b+bU=(1+b)V F b + b U = ( 1 + b ) V
0 ≤ F b − A b ≤ ( 1 + b ) V − D 1 + b V + D b U − b U = o ( V ) ( V ↘ 0 ) . 0\le F_b-A_b\le(1+b)V-D_{1+b}V+D_bU-bU=o(V)\qquad(V\searrow0). 0 ≤ F b − A b ≤ ( 1 + b ) V − D 1 + b V + D b U − b U = o ( V ) ( V ↘ 0 ) . For b ≥ 1 b\ge1 b ≥ 1 b < 1 b<1 b < 1 U ≤ V U\le V U ≤ V
Fix s > b s>b s > b D s U ⊕ A b D_sU\oplus A_b D s U ⊕ A b D s − b U ⊕ D 1 + b V D_{s-b}U\oplus D_{1+b}V D s − b U ⊕ D 1 + b V V V V V V V ( s − b ) U + ( 1 + b ) V + o ( V ) (s-b)U+(1+b)V+o(V) ( s − b ) U + ( 1 + b ) V + o ( V ) V ↘ 0 V\searrow0 V ↘ 0
D s U ⊕ A b = D s − b U ⊕ D 1 + b V . (52) D_sU\oplus A_b=D_{s-b}U\oplus D_{1+b}V. \tag*{(52)} D s U ⊕ A b = D s − b U ⊕ D 1 + b V . (52) To let b ↗ s b\nearrow s b ↗ s b < c b<c b < c
F b ≤ F c ≤ ( 1 + c − b ) F b , A b ≤ A c ≤ ( 1 + c − b ) A b . F_b\le F_c\le(1+c-b)F_b,\qquad A_b\le A_c\le(1+c-b)A_b. F b ≤ F c ≤ ( 1 + c − b ) F b , A b ≤ A c ≤ ( 1 + c − b ) A b . The last inequalities follow by Cauchy comparison and scaling. They also hold for D s U ⊕ A b D_sU\oplus A_b D s U ⊕ A b b ↗ s b\nearrow s b ↗ s D s U ⊕ A s D_sU\oplus A_s D s U ⊕ A s (52) tends to D 1 + s V D_{1+s}V D 1 + s V 1 + b 1+b 1 + b D s − b U D_{s-b}U D s − b U
D s U ⊕ A s = D 1 + s V . D_sU\oplus A_s=D_{1+s}V. D s U ⊕ A s = D 1 + s V . Put α s = s / ( 1 + s ) \alpha_s=s/(1+s) α s = s / ( 1 + s ) W s = D 1 / ( 1 + s ) A s W_s=D_{1/(1+s)}A_s W s = D 1/ ( 1 + s ) A s D α s U ⊕ W s = V D_{\alpha_s}U\oplus W_s=V D α s U ⊕ W s = V W s W_s W s s s s W ≤ V W\le V W ≤ V C \mathcal{C} C V ≤ U ⊕ W s V\le U\oplus W_s V ≤ U ⊕ W s V ≤ U ⊕ W V\le U\oplus W V ≤ U ⊕ W a < 1 a<1 a < 1 s s s D a U ⊕ W ≤ V D_aU\oplus W\le V D a U ⊕ W ≤ V D a ( U ⊕ W ) ≤ V D_a(U\oplus W)\le V D a ( U ⊕ W ) ≤ V a ↗ 1 a\nearrow1 a ↗ 1
Finally q = V − U q=V-U q = V − U f ( V ) − f ( U ) ≤ f ( V − U ) f(V)-f(U)\le f(V-U) f ( V ) − f ( U ) ≤ f ( V − U ) W ~ \widetilde W W q ≤ W ~ ≤ V q\le\widetilde W\le V q ≤ W ≤ V V = U ⊕ W ≤ U + W V=U\oplus W\le U+W V = U ⊕ W ≤ U + W q ≤ W q\le W q ≤ W W ~ ≤ W \widetilde W\le W W ≤ W V ≤ U + W ~ V\le U+\widetilde W V ≤ U + W ⊕ \oplus ⊕ V ≤ U ⊕ W ~ ≤ U ⊕ W = V V\le U\oplus\widetilde W\le U\oplus W=V V ≤ U ⊕ W ≤ U ⊕ W = V W ~ = W \widetilde W=W W = W
Nonlinear subtraction allows us to pass to increasing limits in superposition, as required for the decompositions below.
Corollary 2.17. If U n ↗ U U_n \nearrow U U n ↗ U C \mathcal{C} C V ∈ C V \in\mathcal{C} V ∈ C
U n ⊕ V ↗ U ⊕ V in C l o c 1 , 2 ( R × R N ) as n → ∞ . U_n \oplus V \nearrow U \oplus V \quad\text{in } C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N) \quad\text{as } n \to\infty. U n ⊕ V ↗ U ⊕ V in C loc 1 , 2 ( R × R N ) as n → ∞. Proof. Since U n ⊕ V U_n \oplus V U n ⊕ V n n n 1 1 1 Z Z Z
U n ⊕ V → Z in C l o c 1 , 2 ( R × R N ) as n → ∞ . U_n \oplus V \to Z \quad\text{in } C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N) \quad\text{as } n \to\infty. U n ⊕ V → Z in C loc 1 , 2 ( R × R N ) as n → ∞. Since $0 \le Z \le U \oplus V \in \mathcal{C}$, it belongs to C \mathcal{C} C Z ≥ V Z \ge V Z ≥ V Theorem 2.16 gives a unique X ∈ C X \in\mathcal{C} X ∈ C Z = V ⊕ X Z = V \oplus X Z = V ⊕ X X ≥ U n X \ge U_n X ≥ U n n n n X ≥ U X \ge U X ≥ U Z ≥ U ⊕ V Z \ge U \oplus V Z ≥ U ⊕ V Z = U ⊕ V Z = U \oplus V Z = U ⊕ V
To relate the ordered-space operations to ordinary values of solutions, we compare nonlinear convex combinations with their pointwise averages.
Lemma 2.18. For Z 1 , … , Z m ∈ C Z_1,\ldots,Z_m \in\mathcal{C} Z 1 , … , Z m ∈ C θ j \theta_j θ j ∑ j θ j = 1 \sum_j \theta_j = 1 ∑ j θ j = 1
( ⨁ j = 1 m D θ j Z j ) ( t , x ) ≥ ∑ j = 1 m θ j Z j ( t , x ) . (53) \left(\bigoplus_{j=1}^{m} D_{\theta_j} Z_j\right)(t,x) \ge\sum_{j=1}^{m} \theta_j Z_j(t,x). \tag*{(53)} ( j = 1 ⨁ m D θ j Z j ) ( t , x ) ≥ j = 1 ∑ m θ j Z j ( t , x ) . (53) Proof. First take two terms, with weights θ \theta θ 1 − θ 1-\theta 1 − θ F = θ Z 1 + ( 1 − θ ) Z 2 F = \theta Z_1 + (1-\theta)Z_2 F = θ Z 1 + ( 1 − θ ) Z 2 F ≤ D θ Z 1 + D 1 − θ Z 2 F \le D_\theta Z_1 + D_{1-\theta}Z_2 F ≤ D θ Z 1 + D 1 − θ Z 2 0 < θ < 1 0 < \theta< 1 0 < θ < 1
F ( t ) ≤ S t + T [ F ( − T ) ] ≤ S t + T [ D θ Z 1 ( − T ) + D 1 − θ Z 2 ( − T ) ] . F(t) \le S_{t+T}[F(-T)] \le S_{t+T}[D_\theta Z_1(-T) + D_{1-\theta}Z_2(-T)]. F ( t ) ≤ S t + T [ F ( − T )] ≤ S t + T [ D θ Z 1 ( − T ) + D 1 − θ Z 2 ( − T )] . Let T → ∞ T \to\infty T → ∞
To justify the later small-amplitude analysis, we establish first-order regularity in the amplitude parameter and convergence of the corresponding derivatives.
Lemma 2.19. For every entire solution U U U (1) with 0 < U < 1 0 < U < 1 0 < U < 1 a ↦ D a U a \mapsto D_aU a ↦ D a U C 1 C^1 C 1 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) W = D a U W = D_aU W = D a U
q ( W ) λ ∗ a ≤ ∂ a D a U ≤ W a , ( ∂ t − Δ − f ′ ( W ) ) ∂ a D a U = 0. (54) \frac{q(W)}{\lambda_* a} \le\partial_a D_aU \le\frac{W}{a}, \qquad(\partial_t - \Delta- f'(W))\partial_aD_aU = 0. \tag*{(54)} λ ∗ a q ( W ) ≤ ∂ a D a U ≤ a W , ( ∂ t − Δ − f ′ ( W )) ∂ a D a U = 0. (54) If D a U n → D a U D_aU_n \to D_aU D a U n → D a U a > 0 a > 0 a > 0
∂ a D a U n → ∂ a D a U in C l o c 1 , 2 ( R × R N ) , \partial_aD_aU_n \to\partial_aD_aU \quad\text{in } C_{\mathrm{loc}}^{1,2}(\mathbb{R} \times\mathbb{R}^N), ∂ a D a U n → ∂ a D a U in C loc 1 , 2 ( R × R N ) , locally uniformly in a ∈ ( 0 , ∞ ) a \in(0,\infty) a ∈ ( 0 , ∞ ) S t + T [ a U ( − T ) ] S_{t+T}[aU(-T)] S t + T [ a U ( − T )] T → ∞ T \to\infty T → ∞
V [ U ] : = ∂ a D a U ∣ a = 1 = lim T → ∞ P U ( t , − T ) [ U ( − T ) ] , P U V [ U ] = 0. (55) V[U] := \left.\partial_aD_aU\right|_{a=1} = \lim_{T \to\infty} P_U(t,-T)[U(-T)], \qquad P_UV[U] = 0. \tag*{(55)} V [ U ] := ∂ a D a U ∣ a = 1 = T → ∞ lim P U ( t , − T ) [ U ( − T )] , P U V [ U ] = 0. (55) Proof. At a fixed space–time point, a ↦ D a U a \mapsto D_aU a ↦ D a U a > 0 a > 0 a > 0 W = D a U W = D_aU W = D a U v − v_- v − v + v_+ v +
group law gives D a + h U = D 1 + h / a W D_{a+h}U=D_{1+h/a}W D a + h U = D 1 + h / a W (43) and (44) to W W W h → 0 h\to0 h → 0
q ( W ) λ ∗ a ≤ v + ≤ v − ≤ W a . \frac{q(W)}{\lambda_*a}\le v_+\le v_-\le\frac{W}{a}. λ ∗ a q ( W ) ≤ v + ≤ v − ≤ a W . The last bound follows also from concavity and D 0 U = 0 D_0U=0 D 0 U = 0
For 0 < ∣ h ∣ < a / 2 0<|h|<a/2 0 < ∣ h ∣ < a /2
v h = D a + h U − D a U h v_h=\frac{D_{a+h}U-D_aU}{h} v h = h D a + h U − D a U is nonnegative, bounded by 2 / a 2/a 2/ a
( ∂ t − Δ ) v h = c h v h , c h = ∫ 0 1 f ′ ( W + θ ( D a + h U − W ) ) d θ . (\partial_t-\Delta)v_h=c_hv_h,\qquad c_h=\int_0^1f'\bigl(W+\theta(D_{a+h}U-W)\bigr)\,d\theta. ( ∂ t − Δ ) v h = c h v h , c h = ∫ 0 1 f ′ ( W + θ ( D a + h U − W ) ) d θ . Local convergence c h → f ′ ( W ) c_h\to f'(W) c h → f ′ ( W ) v ± v_\pm v ± P W v ± = 0 P_Wv_\pm=0 P W v ± = 0 P W = ∂ t − Δ − f ′ ( W ) P_W=\partial_t-\Delta-f'(W) P W = ∂ t − Δ − f ′ ( W ) η = v − − v + ≥ 0 \eta=v_--v_+\ge0 η = v − − v + ≥ 0
P W η = 0 , 0 ≤ η ≤ W a E ( W ) , E ( u ) = 1 − q ( u ) λ ∗ u for 0 < u < 1. P_W\eta=0,\qquad0\le\eta\le\frac{W}{a}E(W),\qquad E(u)=1-\frac{q(u)}{\lambda_*u}\quad\text{for }0<u<1. P W η = 0 , 0 ≤ η ≤ a W E ( W ) , E ( u ) = 1 − λ ∗ u q ( u ) for 0 < u < 1. Extend E E E [ 0 , 1 ] [0,1] [ 0 , 1 ] E ( 0 ) = 0 E(0)=0 E ( 0 ) = 0 E ( 1 ) = 1 E(1)=1 E ( 1 ) = 1 P W ( t , s ) P_W(t,s) P W ( t , s ) P W P_W P W T > 0 T>0 T > 0
0 ≤ η ( t ) = 1 T ∫ t − T t P W ( t , s ) [ η ( s ) ] d s ≤ 1 a T ∫ t − T t P W ( t , s ) [ W ( s ) E ( W ( s ) ) ] d s . 0\le\eta(t)=\frac{1}{T}\int_{t-T}^{t}P_W(t,s)[\eta(s)]\,ds\le\frac{1}{aT}\int_{t-T}^{t}P_W(t,s)[W(s)E(W(s))]\,ds. 0 ≤ η ( t ) = T 1 ∫ t − T t P W ( t , s ) [ η ( s )] d s ≤ a T 1 ∫ t − T t P W ( t , s ) [ W ( s ) E ( W ( s ))] d s . Lemma 2.8 makes the last expression tend to zero. Thus v − = v + v_-=v_+ v − = v + (54) .
For differentiable concave functions G n G_n G n
G n ( a + δ ) − G n ( a ) δ ≤ G n ′ ( a ) ≤ G n ( a ) − G n ( a − δ ) δ for 0 < δ < a . \frac{G_n(a+\delta)-G_n(a)}{\delta}\le G_n'(a)\le\frac{G_n(a)-G_n(a-\delta)}{\delta}\qquad\text{for }0<\delta<a. δ G n ( a + δ ) − G n ( a ) ≤ G n ′ ( a ) ≤ δ G n ( a ) − G n ( a − δ ) for 0 < δ < a . If G n → G G_n\to G G n → G G G G n → ∞ n\to\infty n → ∞ δ → 0 \delta\to0 δ → 0 G n ′ ( a ) → G ′ ( a ) G_n'(a)\to G'(a) G n ′ ( a ) → G ′ ( a ) G n ′ G_n' G n ′ G ′ G' G ′ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) G n ( a ) = D a U n ( t , x ) G_n(a)=D_aU_n(t,x) G n ( a ) = D a U n ( t , x ) 0 ≤ ∂ a D a U n ≤ 1 / a 0\le\partial_aD_aU_n\le1/a 0 ≤ ∂ a D a U n ≤ 1/ a (54) , and interior parabolic estimates give
sup a ∈ I ∥ ∂ a D a U n − ∂ a D a U ∥ C 1 , 2 ( K ) → 0 as n → ∞ \sup_{a\in I}\left\|\partial_aD_aU_n-\partial_aD_aU\right\|_{C^{1,2}(K)}\to0\qquad\text{as }n\to\infty a ∈ I sup ∥ ∂ a D a U n − ∂ a D a U ∥ C 1 , 2 ( K ) → 0 as n → ∞ for every compact interval I ⊂ ( 0 , ∞ ) I\subset(0,\infty) I ⊂ ( 0 , ∞ ) K ⊂ R × R N K\subset\mathbb{R}\times\mathbb{R}^N K ⊂ R × R N G n ( a ) = D a + a n − a 0 U ( t , x ) G_n(a)=D_{a+a_n-a_0}U(t,x) G n ( a ) = D a + a n − a 0 U ( t , x ) a n → a 0 > 0 a_n\to a_0>0 a n → a 0 > 0 a ↦ ∂ a D a U a\mapsto\partial_aD_aU a ↦ ∂ a D a U
For u a , T ( t , ⋅ ) = S t + T [ a U ( − T , ⋅ ) ] u_{a,T}(t,\cdot)=S_{t+T}[aU(-T,\cdot)] u a , T ( t , ⋅ ) = S t + T [ a U ( − T , ⋅ )]
0 ≤ ∂ a u a , T ≤ u a , T a ≤ max { a , 1 } a , ( ∂ t − Δ − f ′ ( u a , T ) ) ∂ a u a , T = 0. 0\le\partial_au_{a,T}\le\frac{u_{a,T}}{a}\le\frac{\max\{a,1\}}{a},\qquad(\partial_t-\Delta-f'(u_{a,T}))\partial_au_{a,T}=0. 0 ≤ ∂ a u a , T ≤ a u a , T ≤ a max { a , 1 } , ( ∂ t − Δ − f ′ ( u a , T )) ∂ a u a , T = 0. The same argument gives ∂ a u a , T → ∂ a D a U \partial_au_{a,T}\to\partial_aD_aU ∂ a u a , T → ∂ a D a U C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2 a > 0 a>0 a > 0 T → ∞ T\to\infty T → ∞ a = 1 a=1 a = 1 u 1 , T = U u_{1,T}=U u 1 , T = U
∂ a u a , T ( t , ⋅ ) ∣ a = 1 = P U ( t , − T ) [ U ( − T , ⋅ ) ] , \left.\partial_au_{a,T}(t,\cdot)\right|_{a=1}=P_U(t,-T)[U(-T,\cdot)], ∂ a u a , T ( t , ⋅ ) ∣ a = 1 = P U ( t , − T ) [ U ( − T , ⋅ )] , which proves (55) .
The ordered solution space and translations
Using the algebraic laws and subtraction from Subsection 2.3, we identify C C C E E E E U E_U E U E U E_U E U
Let E E E U ⊖ V U \ominus V U ⊖ V U , V ∈ C U,V \in C U , V ∈ C
U ⊖ V = U ′ ⊖ V ′ ⟺ U ⊕ V ′ = U ′ ⊕ V . U \ominus V = U' \ominus V' \quad\Longleftrightarrow\quad U \oplus V' = U' \oplus V. U ⊖ V = U ′ ⊖ V ′ ⟺ U ⊕ V ′ = U ′ ⊕ V . This rule defines an equivalence relation on the pairs ( U , V ) (U,V) ( U , V ) Proposition 2.11 . Here U ⊖ V U \ominus V U ⊖ V ( U , V ) (U,V) ( U , V )
( U ⊖ V ) ⊕ ( U ′ ⊖ V ′ ) = ( U ⊕ U ′ ) ⊖ ( V ⊕ V ′ ) , a ⊙ ( U ⊖ V ) = D a U ⊖ D a V for a > 0 , ( U ⊖ V ) ⊖ ( U ′ ⊖ V ′ ) = ( U ⊕ V ′ ) ⊖ ( V ⊕ U ′ ) , 0 ⊙ ( U ⊖ V ) = 0 ⊖ 0. (56) \begin{aligned}
(U \ominus V) \oplus(U' \ominus V') &= (U \oplus U') \ominus(V \oplus V'),\\
a \mathbin{\odot} (U \ominus V) &= D_aU \ominus D_aV \quad\text{for } a>0,\\
(U \ominus V) \ominus(U' \ominus V') &= (U \oplus V') \ominus(V \oplus U'),\\
0 \mathbin{\odot} (U \ominus V) &= 0 \ominus0. \tag*{(56)}
\end{aligned} ( U ⊖ V ) ⊕ ( U ′ ⊖ V ′ ) a ⊙ ( U ⊖ V ) ( U ⊖ V ) ⊖ ( U ′ ⊖ V ′ ) 0 ⊙ ( U ⊖ V ) = ( U ⊕ U ′ ) ⊖ ( V ⊕ V ′ ) , = D a U ⊖ D a V for a > 0 , = ( U ⊕ V ′ ) ⊖ ( V ⊕ U ′ ) , = 0 ⊖ 0. (56) For negative scalars use ( − a ) ⊙ x = 0 ⊖ ( a ⊙ x ) (-a) \mathbin{\odot} x=0\ominus(a\mathbin{\odot}x) ( − a ) ⊙ x = 0 ⊖ ( a ⊙ x ) a > 0 a>0 a > 0 x ∈ E x\in E x ∈ E C C C ⊕ \oplus ⊕ D a D_a D a a > 0 a>0 a > 0 E E E Proposition 2.11 ensure that these operations are independent of the representatives and satisfy the vector-space axioms. Thus ( E , ⊕ , ⊙ ) (E,\oplus,\odot) ( E , ⊕ , ⊙ ) 0 ⊖ 0 0\ominus0 0 ⊖ 0 V ⊖ U V\ominus U V ⊖ U U ⊖ V U\ominus V U ⊖ V U ↦ U ⊖ 0 U\mapsto U\ominus0 U ↦ U ⊖ 0 U ⊖ 0 = V ⊖ 0 U\ominus0=V\ominus0 U ⊖ 0 = V ⊖ 0 U ⊕ 0 = V ⊕ 0 U\oplus0=V\oplus0 U ⊕ 0 = V ⊕ 0 U = V U=V U = V
Define its order by
U ⊖ V ≤ U ′ ⊖ V ′ ⟺ U ⊕ V ′ ≤ U ′ ⊕ V . (57) U \ominus V \leq U' \ominus V' \quad\Longleftrightarrow\quad U \oplus V' \leq U' \oplus V. \tag*{(57)} U ⊖ V ≤ U ′ ⊖ V ′ ⟺ U ⊕ V ′ ≤ U ′ ⊕ V . (57) For U , V , U ~ , V ~ , U ′ , V ′ ∈ C U,V,\widetilde{U},\widetilde{V},U',V'\in C U , V , U , V , U ′ , V ′ ∈ C U ⊖ V = U ~ ⊖ V ~ U\ominus V=\widetilde{U}\ominus\widetilde{V} U ⊖ V = U ⊖ V Proposition 2.11 gives
U ⊕ V ′ ≤ U ′ ⊕ V ⟺ U ~ ⊕ V ′ ≤ U ′ ⊕ V ~ . U \oplus V' \leq U' \oplus V \quad\Longleftrightarrow\quad\widetilde{U}\oplus V'\leq U'\oplus\widetilde{V}. U ⊕ V ′ ≤ U ′ ⊕ V ⟺ U ⊕ V ′ ≤ U ′ ⊕ V . The same argument applies when the representation of U ′ ⊖ V ′ U'\ominus V' U ′ ⊖ V ′ E E E E E E U ∈ C U\in C U ∈ C U ⊖ 0 ∈ E U\ominus0\in E U ⊖ 0 ∈ E C C C ⊕ \oplus ⊕ a ⊙ U = D a U a\mathbin{\odot}U=D_aU a ⊙ U = D a U a > 0 a>0 a > 0 ⊕ \oplus ⊕ ⊖ \ominus ⊖ ⊙ \odot ⊙ E E E C C C
By Theorem 2.16 , every positive element U ⊖ V U\ominus V U ⊖ V W ∈ C W\in C W ∈ C V ⊕ W = U V\oplus W=U V ⊕ W = U E E E C C C
An ordered vector space is called a vector lattice if every pair of elements x , y x,y x , y x ∨ y x\vee y x ∨ y x ∧ y x\wedge y x ∧ y E E E
Proposition 2.20. The space E E E n ⊙ x ≤ y n\mathbin{\odot}x\leq y n ⊙ x ≤ y n n n x ≤ 0 x\leq0 x ≤ 0
Proof. For U , V ∈ C U,V\in C U , V ∈ C m ( t , x ) = min { U ( t , x ) , V ( t , x ) } m(t,x)=\min\{U(t,x),V(t,x)\} m ( t , x ) = min { U ( t , x ) , V ( t , x )}
S h [ m ( t ) ] ≤ min { S h [ U ( t ) ] , S h [ V ( t ) ] } = m ( t + h ) . S_h[m(t)]\leq\min\{S_h[U(t)],S_h[V(t)]\}=m(t+h). S h [ m ( t )] ≤ min { S h [ U ( t )] , S h [ V ( t )]} = m ( t + h ) . For R > T R>T R > T − R -R − R − T -T − T t t t
S t + R [ m ( − R ) ] ≤ S t + T [ m ( − T ) ] . S_{t+R}[m(-R)]\leq S_{t+T}[m(-T)]. S t + R [ m ( − R )] ≤ S t + T [ m ( − T )] . Thus the pullbacks decrease. Their limit M M M 0 ≤ M ≤ U , V 0 \le M \le U,V 0 ≤ M ≤ U , V M ∈ C M \in\mathcal{C} M ∈ C Z ∈ C Z \in\mathcal{C} Z ∈ C Z ≤ U , V Z \le U,V Z ≤ U , V − T -T − T Z ( t ) ≤ S t + T [ m ( − T ) ] Z(t) \le S_{t+T}[m(-T)] Z ( t ) ≤ S t + T [ m ( − T )] Z ≤ M Z \le M Z ≤ M M M M U , V U,V U , V U ∧ V U \wedge V U ∧ V
For W ∈ C W \in\mathcal{C} W ∈ C R = ( U ⊕ W ) ∧ ( V ⊕ W ) R = (U \oplus W) \wedge(V \oplus W) R = ( U ⊕ W ) ∧ ( V ⊕ W ) W ≤ R W \le R W ≤ R R = W ⊕ X R = W \oplus X R = W ⊕ X X ∈ C X \in\mathcal{C} X ∈ C X ≤ U , V X \le U,V X ≤ U , V R ≤ W ⊕ ( U ∧ V ) R \le W \oplus(U \wedge V) R ≤ W ⊕ ( U ∧ V ) W ⊕ ( U ∧ V ) W \oplus(U \wedge V) W ⊕ ( U ∧ V ) U ⊕ W U \oplus W U ⊕ W V ⊕ W V \oplus W V ⊕ W
( U ⊕ W ) ∧ ( V ⊕ W ) = ( U ∧ V ) ⊕ W . (58) (U \oplus W) \wedge(V \oplus W) = (U \wedge V) \oplus W. \tag*{(58)} ( U ⊕ W ) ∧ ( V ⊕ W ) = ( U ∧ V ) ⊕ W . (58) This meet is greatest even among signed lower bounds. To see this, let z ∈ E z \in E z ∈ E z ≤ U , V z \le U,V z ≤ U , V W ∈ E + W \in E_+ W ∈ E + z ⊕ W ≥ 0 z \oplus W \ge0 z ⊕ W ≥ 0 W W W E E E
z ⊕ W ≤ ( U ⊕ W ) ∧ ( V ⊕ W ) = ( U ∧ V ) ⊕ W , z \oplus W \le(U \oplus W) \wedge(V \oplus W) = (U \wedge V) \oplus W, z ⊕ W ≤ ( U ⊕ W ) ∧ ( V ⊕ W ) = ( U ∧ V ) ⊕ W , so z ≤ U ∧ V z \le U \wedge V z ≤ U ∧ V
For arbitrary x , y ∈ E x,y \in E x , y ∈ E W ≥ 0 W \ge0 W ≥ 0 x ⊕ W x \oplus W x ⊕ W y ⊕ W y \oplus W y ⊕ W
x ∧ y = ( ( x ⊕ W ) ∧ ( y ⊕ W ) ) ⊖ W . x \wedge y = ((x \oplus W) \wedge(y \oplus W)) \ominus W. x ∧ y = (( x ⊕ W ) ∧ ( y ⊕ W )) ⊖ W . The preceding greatest-lower-bound property shows directly that this is the greatest lower bound of x , y x,y x , y W W W x ∨ y = 0 ⊕ ( ( 0 ⊖ x ) ∧ ( 0 ⊖ y ) ) x \vee y = 0 \oplus((0 \ominus x) \wedge(0 \ominus y)) x ∨ y = 0 ⊕ (( 0 ⊖ x ) ∧ ( 0 ⊖ y )) E E E
To prove the Archimedean property, suppose that n ⊙ x ≤ y n \odot x \le y n ⊙ x ≤ y n n n n ⊙ x + ≤ y + n \odot x^+ \le y^+ n ⊙ x + ≤ y + x + = U x^+ = U x + = U y + = V y^+ = V y + = V C \mathcal{C} C
U ≤ D 1 / n V . U \le D_{1/n}V. U ≤ D 1/ n V . Lemma 2.4 gives U = 0 U = 0 U = 0 n → ∞ n \to\infty n → ∞ x ≤ 0 x \le0 x ≤ 0
Fix a nonzero U ∈ C U \in\mathcal{C} U ∈ C
E U = { x ∈ E : ∣ x ∣ ≤ c ⊙ U for some c ≥ 0 } , ∥ x ∥ U = inf { c ≥ 0 : ∣ x ∣ ≤ c ⊙ U } . E_U = \{x \in E : |x| \le c \odot U \text{ for some } c \ge0\}, \qquad\|x\|_U = \inf\{c \ge0 : |x| \le c \odot U\}. E U = { x ∈ E : ∣ x ∣ ≤ c ⊙ U for some c ≥ 0 } , ∥ x ∥ U = inf { c ≥ 0 : ∣ x ∣ ≤ c ⊙ U } . Here ∣ x ∣ = x ∨ ( 0 ⊖ x ) |x| = x \vee(0 \ominus x) ∣ x ∣ = x ∨ ( 0 ⊖ x ) U U U E U E_U E U x ∈ E U x \in E_U x ∈ E U c > 0 c > 0 c > 0 ( − c ) ⊙ U ≤ x ≤ c ⊙ U (-c) \odot U \le x \le c \odot U ( − c ) ⊙ U ≤ x ≤ c ⊙ U ∥ ⋅ ∥ U \|\cdot\|_U ∥ ⋅ ∥ U
The following proposition proves completeness of the order-unit norm and represents E U E_U E U
Proposition 2.21. The space E U E_U E U ∥ ⋅ ∥ U \|\cdot\|_U ∥ ⋅ ∥ U K U K_U K U J : E U → C ( K U ) J : E_U \to C(K_U) J : E U → C ( K U )
J ( U ) = 1 , ∥ J x ∥ ∞ = ∥ x ∥ U , J ( x ∨ y ) = max { J x , J y } for x , y ∈ E U . J(U) = 1,\qquad\|Jx\|_\infty= \|x\|_U,\qquad J(x \vee y) = \max\{Jx,Jy\}\quad\text{for }x,y \in E_U. J ( U ) = 1 , ∥ J x ∥ ∞ = ∥ x ∥ U , J ( x ∨ y ) = max { J x , J y } for x , y ∈ E U . Proof. For x , y ∈ E U x,y \in E_U x , y ∈ E U α ∈ R \alpha\in\mathbb{R} α ∈ R
∣ α ⊙ x ∣ = ∣ α ∣ ⊙ ∣ x ∣ , ∣ x ⊕ y ∣ ≤ ∣ x ∣ ⊕ ∣ y ∣ |\alpha\odot x| = |\alpha| \odot|x|,\qquad|x \oplus y| \le|x| \oplus|y| ∣ α ⊙ x ∣ = ∣ α ∣ ⊙ ∣ x ∣ , ∣ x ⊕ y ∣ ≤ ∣ x ∣ ⊕ ∣ y ∣ give homogeneity and the triangle inequality. By Proposition 2.20,
∥ x ∥ U ≤ c ⟹ ∣ x ∣ ≤ ( c + n − 1 ) ⊙ U for every n ≥ 1 ⟹ ∣ x ∣ ≤ c ⊙ U . \|x\|_U \le c \Longrightarrow|x| \le(c+n^{-1}) \odot U\quad\text{for every }n \ge1 \Longrightarrow|x| \le c \odot U. ∥ x ∥ U ≤ c ⟹ ∣ x ∣ ≤ ( c + n − 1 ) ⊙ U for every n ≥ 1 ⟹ ∣ x ∣ ≤ c ⊙ U . Thus ∥ ⋅ ∥ U \lVert\cdot\rVert_U ∥ ⋅ ∥ U
∣ x ∣ ≤ ∣ y ∣ ⟹ ∥ x ∥ U ≤ ∥ y ∥ U , |x|\le|y|\quad\Longrightarrow\quad\lVert x\rVert_U\le\lVert y\rVert_U, ∣ x ∣ ≤ ∣ y ∣ ⟹ ∥ x ∥ U ≤ ∥ y ∥ U , ∥ x ∨ y ∥ U = max { ∥ x ∥ U , ∥ y ∥ U } for x , y ≥ 0. \lVert x\vee y\rVert_U=\max\{\lVert x\rVert_U,\lVert y\rVert_U\}\qquad\text{for }x,y\ge0. ∥ x ∨ y ∥ U = max {∥ x ∥ U , ∥ y ∥ U } for x , y ≥ 0. The second identity follows from x ∨ y ≤ c ⊙ U ⟺ x , y ≤ c ⊙ U x\vee y\le c\odot U\Longleftrightarrow x,y\le c\odot U x ∨ y ≤ c ⊙ U ⟺ x , y ≤ c ⊙ U
If 0 ≤ A n ↗ 0\le A_n\nearrow 0 ≤ A n ↗ A n ≤ c ⊙ U A_n\le c\odot U A n ≤ c ⊙ U c > 0 c>0 c > 0
A n ⟶ A in C l o c 1 , 2 ( R × R N ) , 0 ≤ A ≤ D c U < 1 , A ∈ C ∩ E U . A_n\longrightarrow A\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N),\qquad0\le A\le D_cU<1,\qquad A\in\mathcal{C}\cap E_U. A n ⟶ A in C loc 1 , 2 ( R × R N ) , 0 ≤ A ≤ D c U < 1 , A ∈ C ∩ E U . For every upper bound B ∈ E B\in E B ∈ E
B ≥ A n ≥ 0 for all n ⟹ B ∈ C , B ≥ A . B\ge A_n\ge0\quad\text{for all }n\quad\Longrightarrow\quad B\in\mathcal{C},\quad B\ge A. B ≥ A n ≥ 0 for all n ⟹ B ∈ C , B ≥ A . Hence A = sup n A n A=\sup_n A_n A = sup n A n E E E c = 0 c=0 c = 0
Let { x n } n ≥ 1 ⊂ E U \{x_n\}_{n\ge1}\subset E_U { x n } n ≥ 1 ⊂ E U d j = x n j + 1 ⊖ x n j d_j=x_{n_{j+1}}\ominus x_{n_j} d j = x n j + 1 ⊖ x n j
∣ d j ∣ ≤ 2 − j ⊙ U , P m : = ⨁ j = 1 m d j + , N m : = ⨁ j = 1 m d j − . |d_j|\le2^{-j}\odot U,\qquad P_m:=\bigoplus_{j=1}^{m}d_j^+,\qquad N_m:=\bigoplus_{j=1}^{m}d_j^-. ∣ d j ∣ ≤ 2 − j ⊙ U , P m := j = 1 ⨁ m d j + , N m := j = 1 ⨁ m d j − . Then 0 ≤ P m , N m ↗ ≤ U 0\le P_m,N_m\nearrow\le U 0 ≤ P m , N m ↗≤ U P = sup m P m P=\sup_m P_m P = sup m P m N = sup m N m N=\sup_m N_m N = sup m N m E U E_U E U k ≥ m k\ge m k ≥ m
0 ≤ P k ⊖ P m , N k ⊖ N m ≤ ( ∑ j = m + 1 k 2 − j ) ⊙ U ≤ 2 − m ⊙ U . 0\le P_k\ominus P_m,\ N_k\ominus N_m\le\left(\sum_{j=m+1}^{k}2^{-j}\right)\odot U\le2^{-m}\odot U. 0 ≤ P k ⊖ P m , N k ⊖ N m ≤ ( j = m + 1 ∑ k 2 − j ) ⊙ U ≤ 2 − m ⊙ U . Taking suprema yields
0 ≤ P ⊖ P m , N ⊖ N m ≤ 2 − m ⊙ U . 0\le P\ominus P_m,\ N\ominus N_m\le2^{-m}\odot U. 0 ≤ P ⊖ P m , N ⊖ N m ≤ 2 − m ⊙ U . Since x n m + 1 = x n 1 ⊕ P m ⊖ N m x_{n_{m+1}}=x_{n_1}\oplus P_m\ominus N_m x n m + 1 = x n 1 ⊕ P m ⊖ N m
∥ x n m + 1 ⊖ ( x n 1 ⊕ P ⊖ N ) ∥ U ≤ 2 1 − m ⟶ 0 as m → ∞ . \lVert x_{n_{m+1}}\ominus(x_{n_1}\oplus P\ominus N)\rVert_U\le2^{1-m}\longrightarrow0\qquad\text{as }m\to\infty. ∥ x n m + 1 ⊖ ( x n 1 ⊕ P ⊖ N ) ∥ U ≤ 2 1 − m ⟶ 0 as m → ∞. The Cauchy property therefore gives x n → x n 1 ⊕ P ⊖ N x_n\to x_{n_1}\oplus P\ominus N x n → x n 1 ⊕ P ⊖ N E U E_U E U
Finally,
{ x ∈ E U : ∥ x ∥ U ≤ 1 } = [ ( − 1 ) ⊙ U , U ] . \{x\in E_U:\lVert x\rVert_U\le1\}=[(-1)\odot U,U]. { x ∈ E U : ∥ x ∥ U ≤ 1 } = [( − 1 ) ⊙ U , U ] . Kakutani’s representation theorem [30 ] , applied with the norm properties established above, gives K U K_U K U J J J
The norm constructed above must also control convergence of the physical solutions. The next lemma connects the order-unit norm to local parabolic convergence.
Lemma 2.22. If W , V ∈ E U ∩ C W,V\in E_U\cap\mathcal{C} W , V ∈ E U ∩ C ∥ W ⊖ V ∥ U ≤ ε \lVert W\ominus V\rVert_U\le\varepsilon ∥ W ⊖ V ∥ U ≤ ε ε > 0 \varepsilon>0 ε > 0
∣ W ( t , x ) − V ( t , x ) ∣ ≤ D ε U ( t , x ) . |W(t,x)-V(t,x)|\le D_\varepsilon U(t,x). ∣ W ( t , x ) − V ( t , x ) ∣ ≤ D ε U ( t , x ) . In particular, for V n , V ∈ E U ∩ C V_n,V\in E_U\cap\mathcal{C} V n , V ∈ E U ∩ C
∥ V n ⊖ V ∥ U ⟶ 0 ⟹ V n ⟶ V in C l o c 1 , 2 ( R × R N ) as n → ∞ . \lVert V_n\ominus V\rVert_U\longrightarrow0\quad\Longrightarrow\quad V_n\longrightarrow V\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N)\qquad\text{as }n\to\infty. ∥ V n ⊖ V ∥ U ⟶ 0 ⟹ V n ⟶ V in C loc 1 , 2 ( R × R N ) as n → ∞. Proof. The norm inequality gives W ≤ V ⊕ D ε U W\le V\oplus D_\varepsilon U W ≤ V ⊕ D ε U V ≤ W ⊕ D ε U V\le W\oplus D_\varepsilon U V ≤ W ⊕ D ε U E E E
W ≤ V ⊕ D ε U ≤ V + D ε U , V ≤ W ⊕ D ε U ≤ W + D ε U , W\le V\oplus D_\varepsilon U\le V+D_\varepsilon U,\qquad V\le W\oplus D_\varepsilon U\le W+D_\varepsilon U, W ≤ V ⊕ D ε U ≤ V + D ε U , V ≤ W ⊕ D ε U ≤ W + D ε U , where the final sums are ordinary sums of functions. This proves the pointwise bound. Lemma 2.4 and parabolic compactness prove the convergence assertion.
We now study space–time translations on the ordered solution space and on its principal ideals. For s ∈ R s \in\mathbb{R} s ∈ R h ∈ R N h \in\mathbb{R}^{N} h ∈ R N
( T s , h V ) ( t , x ) = V ( t + s , x + h ) for V ∈ C . (T_{s,h}V)(t,x)=V(t+s,x+h) \qquad\text{for } V\in\mathcal{C}. ( T s , h V ) ( t , x ) = V ( t + s , x + h ) for V ∈ C . Translations preserve superposition and dilation, so they extend to linear order isomorphisms of E E E T − s , − h T_{-s,-h} T − s , − h
The differential estimates now give order bounds for translations. These bounds show that translations preserve the subspaces E U E_U E U
Lemma 2.23. The following operator inequalities hold on E + = C E_+=\mathcal{C} E + = C
e − λ ∗ ∣ h ∣ I ≤ T 0 , h ≤ e λ ∗ ∣ h ∣ I , e^{-\lambda_*|h|}I\leq T_{0,h}\leq e^{\lambda_*|h|}I, e − λ ∗ ∣ h ∣ I ≤ T 0 , h ≤ e λ ∗ ∣ h ∣ I , and, for s ≥ 0 s\geq0 s ≥ 0
I ≤ T s , 0 ≤ e 2 f ′ ( 0 ) s I . I\leq T_{s,0}\leq e^{2f'(0)s}I. I ≤ T s , 0 ≤ e 2 f ′ ( 0 ) s I . Every T s , h T_{s,h} T s , h E U E_U E U
Proof. The bounds (36) imply the ordinary pointwise inequalities
e − λ ∗ ∣ h ∣ V ( t , x ) ≤ V ( t , x + h ) ≤ e λ ∗ ∣ h ∣ V ( t , x ) , e^{-\lambda_*|h|}V(t,x)\leq V(t,x+h)\leq e^{\lambda_*|h|}V(t,x), e − λ ∗ ∣ h ∣ V ( t , x ) ≤ V ( t , x + h ) ≤ e λ ∗ ∣ h ∣ V ( t , x ) , V ( t , x ) ≤ V ( t + s , x ) ≤ e 2 f ′ ( 0 ) s V ( t , x ) for s ≥ 0. V(t,x)\leq V(t+s,x)\leq e^{2f'(0)s}V(t,x)\qquad\text{for }s\geq0. V ( t , x ) ≤ V ( t + s , x ) ≤ e 2 f ′ ( 0 ) s V ( t , x ) for s ≥ 0. To convert these into inequalities for vector-space scalar multiplication, observe that if entire solutions Z , V Z,V Z , V Z ≤ a V Z\leq aV Z ≤ aV − T -T − T
Z ( t ) ≤ S t + T [ a V ( − T ) ] , Z(t)\leq S_{t+T}[aV(-T)], Z ( t ) ≤ S t + T [ aV ( − T )] , and hence Z ≤ D a V Z\leq D_aV Z ≤ D a V Z ≥ D a V Z\geq D_aV Z ≥ D a V s s s e 2 f ′ ( 0 ) s I ≤ T s , 0 ≤ I e^{2f'(0)s}I\leq T_{s,0}\leq I e 2 f ′ ( 0 ) s I ≤ T s , 0 ≤ I s , h s,h s , h
e − 2 f ′ ( 0 ) ∣ s ∣ − λ ∗ ∣ h ∣ I ≤ T s , h ≤ e 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣ I . e^{-2f'(0)|s|-\lambda_*|h|}I\leq T_{s,h}\leq e^{2f'(0)|s|+\lambda_*|h|}I. e − 2 f ′ ( 0 ) ∣ s ∣ − λ ∗ ∣ h ∣ I ≤ T s , h ≤ e 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣ I . If V ∈ E U V\in E_U V ∈ E U ∣ V ∣ ≤ c ⊙ U |V|\leq c\mathbin{\odot}U ∣ V ∣ ≤ c ⊙ U T s , h T_{s,h} T s , h
∣ T s , h V ∣ = T s , h ∣ V ∣ ≤ c ⊙ T s , h U ≤ ( c e 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣ ) ⊙ U . |T_{s,h}V|=T_{s,h}|V|\leq c\mathbin{\odot}T_{s,h}U\leq\left(ce^{2f'(0)|s|+\lambda_*|h|}\right)\mathbin{\odot}U. ∣ T s , h V ∣ = T s , h ∣ V ∣ ≤ c ⊙ T s , h U ≤ ( c e 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣ ) ⊙ U . Thus T s , h T_{s,h} T s , h E U E_U E U
In this representation, space–time translations become multiplication by positive continuous functions. We identify their exponential form.
Proposition 2.24. For the representation J : E U → C ( K U ) J:E_U\to C(K_U) J : E U → C ( K U )
λ : K U → [ 0 , 2 f ′ ( 0 ) ] , ξ : K U → B ‾ 1 ⊂ R N \lambda:K_U\to[0,2f'(0)],\qquad\xi:K_U\to\overline{B}_1\subset\mathbb{R}^{N} λ : K U → [ 0 , 2 f ′ ( 0 )] , ξ : K U → B 1 ⊂ R N such that
J ( T s , h V ) ( k ) = e s λ ( k ) + λ ∗ h ⋅ ξ ( k ) J ( V ) ( k ) (59) J(T_{s,h}V)(k)=e^{s\lambda(k)+\lambda_*h\cdot\xi(k)}J(V)(k) \tag*{(59)} J ( T s , h V ) ( k ) = e s λ ( k ) + λ ∗ h ⋅ ξ ( k ) J ( V ) ( k ) (59) for every V ∈ E U V\in E_U V ∈ E U k ∈ K U k\in K_U k ∈ K U ( s , h ) ∈ R 1 + N (s,h)\in\mathbb{R}^{1+N} ( s , h ) ∈ R 1 + N
∥ T s , h − I ∥ E U → E U ≤ e 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣ − 1. \|T_{s,h}-I\|_{E_U\to E_U}\leq e^{2f'(0)|s|+\lambda_*|h|}-1. ∥ T s , h − I ∥ E U → E U ≤ e 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣ − 1. Proof. We first prove an elementary fact about positive operators. Let T : C ( K U ) → C ( K U ) T:C(K_U)\to C(K_U) T : C ( K U ) → C ( K U ) 0 ≤ T ≤ β I 0\le T\le\beta I 0 ≤ T ≤ β I f ∈ C ( K U ) f\in C(K_U) f ∈ C ( K U )
∣ T f ∣ ≤ T ∣ f ∣ ≤ β ∣ f ∣ . |Tf|\le T|f|\le\beta|f|. ∣ T f ∣ ≤ T ∣ f ∣ ≤ β ∣ f ∣. Fix k ∈ K U k\in K_U k ∈ K U f ( k ) = 0 f(k)=0 f ( k ) = 0 ( T f ) ( k ) = 0 (Tf)(k)=0 ( T f ) ( k ) = 0 f − f ( k ) 1 f-f(k)1 f − f ( k ) 1
( T f ) ( k ) = g ( k ) f ( k ) , g = T 1 ∈ C ( K U ) . (Tf)(k)=g(k)f(k),\qquad g=T1\in C(K_U). ( T f ) ( k ) = g ( k ) f ( k ) , g = T 1 ∈ C ( K U ) . Thus T T T g g g T ≥ α I T\ge\alpha I T ≥ α I α ≤ g ≤ β \alpha\le g\le\beta α ≤ g ≤ β
Transport each translation to C ( K U ) C(K_U) C ( K U ) Lemma 2.23 . Its multiplier g s , h g_{s,h} g s , h k k k
g s + r , h + ℓ = g s , h g r , ℓ , ∣ ln g s , h ( k ) ∣ ≤ 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣ . g_{s+r,h+\ell}=g_{s,h}g_{r,\ell},\qquad|\ln g_{s,h}(k)|\le2f'(0)|s|+\lambda_*|h|. g s + r , h + ℓ = g s , h g r , ℓ , ∣ ln g s , h ( k ) ∣ ≤ 2 f ′ ( 0 ) ∣ s ∣ + λ ∗ ∣ h ∣. For a fixed k k k ( s , h ) (s,h) ( s , h )
ln g s , h ( k ) = s λ ( k ) + λ ∗ h ⋅ ξ ( k ) . \ln g_{s,h}(k)=s\lambda(k)+\lambda_*h\cdot\xi(k). ln g s , h ( k ) = s λ ( k ) + λ ∗ h ⋅ ξ ( k ) . One may define λ ( k ) = ln g 1 , 0 ( k ) \lambda(k)=\ln g_{1,0}(k) λ ( k ) = ln g 1 , 0 ( k ) ξ i ( k ) = λ ∗ − 1 ln g 0 , e i ( k ) \xi_i(k)=\lambda_*^{-1}\ln g_{0,e_i}(k) ξ i ( k ) = λ ∗ − 1 ln g 0 , e i ( k ) k k k 1 ≤ g s , 0 ≤ e 2 f ′ ( 0 ) s 1\le g_{s,0}\le e^{2f'(0)s} 1 ≤ g s , 0 ≤ e 2 f ′ ( 0 ) s s ≥ 0 s\ge0 s ≥ 0 0 ≤ λ ≤ 2 f ′ ( 0 ) 0\le\lambda\le2f'(0) 0 ≤ λ ≤ 2 f ′ ( 0 ) e − λ ∗ ∣ h ∣ ≤ g 0 , h ≤ e λ ∗ ∣ h ∣ e^{-\lambda_*|h|}\le g_{0,h}\le e^{\lambda_*|h|} e − λ ∗ ∣ h ∣ ≤ g 0 , h ≤ e λ ∗ ∣ h ∣ ∣ λ ∗ h ⋅ ξ ( k ) ∣ ≤ λ ∗ ∣ h ∣ |\lambda_*h\cdot\xi(k)|\le\lambda_*|h| ∣ λ ∗ h ⋅ ξ ( k ) ∣ ≤ λ ∗ ∣ h ∣ h h h ∣ ξ ( k ) ∣ ≤ 1 |\xi(k)|\le1 ∣ ξ ( k ) ∣ ≤ 1 (59) . Finally, the norm of a multiplication operator on C ( K U ) C(K_U) C ( K U )
Linear representations and small-amplitude limits
This section relates nonlinear entire solutions to nonnegative entire solutions of the linearized equation and their representing measures. We first assume the finiteness condition (68) . Under this condition, linear evolution of the past data U ( − T , ⋅ ) U(-T,\cdot) U ( − T , ⋅ ) H H H T → ∞ T\to\infty T → ∞ H H H U U U U U U
We then establish a sufficient condition for this reconstruction. If ∣ ∇ ln U ∣ ≤ λ ∗ r |\nabla\ln U|\le\lambda_*r ∣∇ ln U ∣ ≤ λ ∗ r 0 ≤ r < 1 0\le r<1 0 ≤ r < 1 D a U / a D_aU/a D a U / a B ‾ r \overline{B}_r B r
Linear reconstruction and finite-front approximation
We prove reconstruction and finite HN approximation under (68) . The preliminary estimates treat capped critical exponential fields and finite families of fronts; they will be used to control the reconstruction and identify its finite atomic approximants.
Throughout the section, f f f (2) with f ′ ( 0 ) > 0 f'(0)>0 f ′ ( 0 ) > 0 S τ f = S τ S_\tau^f=S_\tau S τ f = S τ f f f V h = V h f V_h=V_h^f V h = V h f
Concavity implies 0 ≤ f ( u ) ≤ f ′ ( 0 ) u 0\le f(u)\le f'(0)u 0 ≤ f ( u ) ≤ f ′ ( 0 ) u f ′ ( u ) ≤ f ′ ( 0 ) f'(u)\le f'(0) f ′ ( u ) ≤ f ′ ( 0 ) F 1 F_1 F 1 (46) , Taylor’s formula gives
0 ≤ F 1 ( u ) ≤ M u 2 for 0 ≤ u ≤ 1 , M = 1 2 ∥ f ′ ′ ∥ L ∞ ( [ 0 , 1 ] ) . (60) 0\le F_1(u)\le Mu^2\quad\text{for }0\le u\le1,\qquad M=\frac{1}{2}\lVert f''\rVert_{L^\infty([0,1])}. \tag*{(60)} 0 ≤ F 1 ( u ) ≤ M u 2 for 0 ≤ u ≤ 1 , M = 2 1 ∥ f ′′ ∥ L ∞ ([ 0 , 1 ]) . (60) In particular M ≥ f ′ ( 0 ) M \ge f'(0) M ≥ f ′ ( 0 ) f ′ ( 0 ) − f ′ ( 1 ) = f ′ ( 0 ) f'(0)-f'(1)=f'(0) f ′ ( 0 ) − f ′ ( 1 ) = f ′ ( 0 )
∣ S τ f [ a ] − S τ f [ b ] ∣ ≤ L τ [ ∣ a − b ∣ ] , 0 ≤ a , b ≤ 1. (61) \left|S_\tau^f[a]-S_\tau^f[b]\right|\le L_\tau[|a-b|],\qquad0\le a,b\le1. \tag*{(61)} S τ f [ a ] − S τ f [ b ] ≤ L τ [ ∣ a − b ∣ ] , 0 ≤ a , b ≤ 1. (61) The concave extension fixed after (2) is used if a sum exceeds one. Lemma 2.2 and comparison give
S τ f [ min { a + b , 1 } ] ≤ S τ f [ a ] + S τ f [ b ] . (62) S_\tau^f[\min\{a+b,1\}]\le S_\tau^f[a]+S_\tau^f[b]. \tag*{(62)} S τ f [ min { a + b , 1 }] ≤ S τ f [ a ] + S τ f [ b ] . (62) We first consider linear solutions generated by finite measures on the critical sphere. Let μ \mu μ S N − 1 S^{N-1} S N − 1
H μ ( t , x ) = e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ ν ⋅ x d μ ( ν ) = ∫ S N − 1 e λ ∗ ν ⋅ x + 2 f ′ ( 0 ) t d μ ( ν ) . (63) H_\mu(t,x)=e^{2f'(0)t}\int_{S^{N-1}}e^{\lambda^*\nu\cdot x}\,d\mu(\nu)=\int_{S^{N-1}}e^{\lambda^*\nu\cdot x+2f'(0)t}\,d\mu(\nu). \tag*{(63)} H μ ( t , x ) = e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ ν ⋅ x d μ ( ν ) = ∫ S N − 1 e λ ∗ ν ⋅ x + 2 f ′ ( 0 ) t d μ ( ν ) . (63) For each ν ∈ S N − 1 \nu\in S^{N-1} ν ∈ S N − 1 u t = Δ u + f ′ ( 0 ) u u_t=\Delta u+f'(0)u u t = Δ u + f ′ ( 0 ) u c ∗ c_* c ∗ − ν -\nu − ν H μ H_\mu H μ
Let L t = e f ′ ( 0 ) t P t L_t=e^{f'(0)t}P_t L t = e f ′ ( 0 ) t P t u t = Δ u + f ′ ( 0 ) u u_t=\Delta u+f'(0)u u t = Δ u + f ′ ( 0 ) u ( L t + n [ a H μ ( − n , ⋅ ) ] ) ( x ) = a H μ ( t , x ) (L_{t+n}[aH_\mu(-n,\cdot)])(x)=aH_\mu(t,x) ( L t + n [ a H μ ( − n , ⋅ )]) ( x ) = a H μ ( t , x ) t + n > 0 t+n>0 t + n > 0
Lemma 3.1. For every a > 0 a>0 a > 0 ( t , x ) ∈ R × R N (t,x)\in\mathbb{R}\times\mathbb{R}^N ( t , x ) ∈ R × R N
( L t + n [ min { a H μ ( − n , ⋅ ) , 1 } ] ) ( x ) → a 2 H μ ( t , x ) as n ∈ N + and n → + ∞ . \left(L_{t+n}[\min\{aH_\mu(-n,\cdot),1\}]\right)(x)\to\frac{a}{2}H_\mu(t,x)\qquad\text{as }n\in\mathbb{N}_+\text{ and }n\to+\infty. ( L t + n [ min { a H μ ( − n , ⋅ ) , 1 }] ) ( x ) → 2 a H μ ( t , x ) as n ∈ N + and n → + ∞. Proof. The result is immediate if μ = 0 \mu=0 μ = 0 M = μ ( S N − 1 ) > 0 M=\mu(S^{N-1})>0 M = μ ( S N − 1 ) > 0 μ \mu μ ν \nu ν
ln μ ( B ( ν , n − 1 / 2 ) ) n → 0. (64) \frac{\ln\mu(B(\nu,n^{-1/2}))}{\sqrt{n}}\to0. \tag*{(64)} n ln μ ( B ( ν , n − 1/2 )) → 0. (64) Equivalently, there exists a Borel set E ⊂ S N − 1 E\subset S^{N-1} E ⊂ S N − 1 μ ( E ) = 0 \mu(E)=0 μ ( E ) = 0 (64) holds for every ν ∈ S N − 1 ∖ E \nu\in S^{N-1}\setminus E ν ∈ S N − 1 ∖ E
For each n ≥ 1 n\ge1 n ≥ 1 1 / ( 2 N n ) 1/(2\sqrt{Nn}) 1/ ( 2 N n ) Q n , 1 , … , Q n , K n Q_{n,1},\ldots,Q_{n,K_n} Q n , 1 , … , Q n , K n
K n ≤ C N N / 2 , diam Q n , j ≤ 1 2 n < n − 1 / 2 . K_n\le CN^{N/2},\qquad\operatorname{diam}Q_{n,j}\le\frac{1}{2\sqrt{n}}<n^{-1/2}. K n ≤ C N N /2 , diam Q n , j ≤ 2 n 1 < n − 1/2 . Fix ε > 0 \varepsilon>0 ε > 0
E n , ε : = { ν ∈ S N − 1 : μ ( B ( ν , n − 1 / 2 ) ) < e − ε n } . E_{n,\varepsilon}:=\left\{\nu\in S^{N-1}:\mu(B(\nu,n^{-1/2}))<e^{-\varepsilon\sqrt{n}}\right\}. E n , ε := { ν ∈ S N − 1 : μ ( B ( ν , n − 1/2 )) < e − ε n } . If Q n , j ∩ E n , ε ≠ ∅ Q_{n,j}\cap E_{n,\varepsilon}\ne\varnothing Q n , j ∩ E n , ε = ∅ ν ∈ Q n , j ∩ E n , ε \nu\in Q_{n,j}\cap E_{n,\varepsilon} ν ∈ Q n , j ∩ E n , ε
Q n , j ⊂ B ( ν , n − 1 / 2 ) , μ ( Q n , j ) ≤ μ ( B ( ν , n − 1 / 2 ) ) < e − ε n . Q_{n,j}\subset B(\nu,n^{-1/2}),\qquad\mu(Q_{n,j})\le\mu(B(\nu,n^{-1/2}))<e^{-\varepsilon\sqrt{n}}. Q n , j ⊂ B ( ν , n − 1/2 ) , μ ( Q n , j ) ≤ μ ( B ( ν , n − 1/2 )) < e − ε n . Summing over the partition cells that meet E n , ε E_{n,\varepsilon} E n , ε
μ ( E n , ε ) ≤ ∑ j : Q n , j ∩ E n , ε ≠ ∅ μ ( Q n , j ) ≤ C N N / 2 e − ε n . \mu(E_{n,\varepsilon})\le\sum_{\substack{j:Q_{n,j}\cap E_{n,\varepsilon}\ne\varnothing}}\mu(Q_{n,j})\le CN^{N/2}e^{-\varepsilon\sqrt{n}}. μ ( E n , ε ) ≤ j : Q n , j ∩ E n , ε = ∅ ∑ μ ( Q n , j ) ≤ C N N /2 e − ε n . In particular, ∑ n = 1 ∞ μ ( E n , ε ) < ∞ \sum_{n=1}^{\infty}\mu(E_{n,\varepsilon})<\infty ∑ n = 1 ∞ μ ( E n , ε ) < ∞
The set of points belonging to infinitely many E n , ε E_{n,\varepsilon} E n , ε
E ε : = ⋂ m = 1 ∞ ⋃ n = m ∞ E n , ε . E_{\varepsilon}:=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}E_{n,\varepsilon}. E ε := m = 1 ⋂ ∞ n = m ⋃ ∞ E n , ε . For every m ≥ 1 m\ge1 m ≥ 1
μ ( E ε ) ≤ ∑ n = m ∞ μ ( E n , ε ) → 0 ( m → ∞ ) , \mu(E_{\varepsilon})\le\sum_{n=m}^{\infty}\mu(E_{n,\varepsilon})\to0\qquad(m\to\infty), μ ( E ε ) ≤ n = m ∑ ∞ μ ( E n , ε ) → 0 ( m → ∞ ) , so μ ( E ε ) = 0 \mu(E_{\varepsilon})=0 μ ( E ε ) = 0 E : = ⋃ k = 1 ∞ E 1 / k E:=\bigcup_{k=1}^{\infty}E_{1/k} E := ⋃ k = 1 ∞ E 1/ k μ ( E ) = 0 \mu(E)=0 μ ( E ) = 0 ν ∉ E \nu\notin E ν ∈ / E k ≥ 1 k\ge1 k ≥ 1 n 0 = n 0 ( ν , k ) n_0=n_0(\nu,k) n 0 = n 0 ( ν , k ) ν ∉ E n , 1 / k \nu\notin E_{n,1/k} ν ∈ / E n , 1/ k n ≥ n 0 n\ge n_0 n ≥ n 0 μ ( B ( ν , n − 1 / 2 ) ) ≤ M \mu(B(\nu,n^{-1/2}))\le M μ ( B ( ν , n − 1/2 )) ≤ M
− 1 k ≤ ln μ ( B ( ν , n − 1 / 2 ) ) n ≤ ln M n for n ≥ n 0 . -\frac{1}{k}\le\frac{\ln\mu(B(\nu,n^{-1/2}))}{\sqrt{n}}\le\frac{\ln M}{\sqrt{n}}\qquad\text{for }n\ge n_0. − k 1 ≤ n ln μ ( B ( ν , n − 1/2 )) ≤ n ln M for n ≥ n 0 . Letting first n → ∞ n\to\infty n → ∞ k → ∞ k\to\infty k → ∞ (64) for every ν ∉ E \nu\notin E ν ∈ / E
Fix a > 0 a>0 a > 0 t ∈ R t\in\mathbb{R} t ∈ R ν ∈ S N − 1 ∖ E \nu\in S^{N-1}\setminus E ν ∈ S N − 1 ∖ E x , z ∈ R N x,z\in\mathbb{R}^{N} x , z ∈ R N y n = x + 2 λ ∗ ( t + n ) ν + t + n z y_n=x+2\lambda_*(t+n)\nu+\sqrt{t+n}\,z y n = x + 2 λ ∗ ( t + n ) ν + t + n z
ln ( a H μ ( − n , y n ) ) t + n ⟶ λ ∗ ν ⋅ z as n → ∞ . (65) \frac{\ln(aH_\mu(-n,y_n))}{\sqrt{t+n}}\longrightarrow\lambda_*\nu\cdot z\qquad\text{as }n\to\infty. \tag*{(65)} t + n ln ( a H μ ( − n , y n )) ⟶ λ ∗ ν ⋅ z as n → ∞. (65) For the upper bound, H μ ( − n , y n ) ≤ M e − 2 f ′ ( 0 ) n + λ ∗ ∣ y n ∣ H_\mu(-n,y_n)\le Me^{-2f'(0)n+\lambda_*|y_n|} H μ ( − n , y n ) ≤ M e − 2 f ′ ( 0 ) n + λ ∗ ∣ y n ∣ ∣ y n ∣ = 2 λ ∗ ( t + n ) + t + n ν ⋅ z + O x , z ( 1 ) |y_n|=2\lambda_*(t+n)+\sqrt{t+n}\,\nu\cdot z+O_{x,z}(1) ∣ y n ∣ = 2 λ ∗ ( t + n ) + t + n ν ⋅ z + O x , z ( 1 ) H μ H_\mu H μ ∣ ζ − ν ∣ < n − 1 / 2 |\zeta-\nu|<n^{-1/2} ∣ ζ − ν ∣ < n − 1/2 n n n 0 < t + n ≤ 2 n 0<t+n\le2n 0 < t + n ≤ 2 n
2 f ′ ( 0 ) ( t + n ) ν ⋅ ζ = 2 f ′ ( 0 ) ( t + n ) − f ′ ( 0 ) ( t + n ) ∣ ν − ζ ∣ 2 ≥ 2 f ′ ( 0 ) ( t + n ) − 2 f ′ ( 0 ) , 2f'(0)(t+n)\nu\cdot\zeta=2f'(0)(t+n)-f'(0)(t+n)|\nu-\zeta|^2\ge2f'(0)(t+n)-2f'(0), 2 f ′ ( 0 ) ( t + n ) ν ⋅ ζ = 2 f ′ ( 0 ) ( t + n ) − f ′ ( 0 ) ( t + n ) ∣ ν − ζ ∣ 2 ≥ 2 f ′ ( 0 ) ( t + n ) − 2 f ′ ( 0 ) , λ ∗ t + n ζ ⋅ z ≥ λ ∗ t + n ν ⋅ z − 2 λ ∗ ∣ z ∣ , λ ∗ ζ ⋅ x ≥ − λ ∗ ∣ x ∣ . \lambda_*\sqrt{t+n}\,\zeta\cdot z\ge\lambda_*\sqrt{t+n}\,\nu\cdot z-\sqrt{2}\lambda_*|z|,\qquad\lambda_*\zeta\cdot x\ge-\lambda_*|x|. λ ∗ t + n ζ ⋅ z ≥ λ ∗ t + n ν ⋅ z − 2 λ ∗ ∣ z ∣ , λ ∗ ζ ⋅ x ≥ − λ ∗ ∣ x ∣. This gives
H μ ( − n , y n ) ≥ μ ( B ( ν , n − 1 / 2 ) ) e 2 f ′ ( 0 ) t + λ ∗ t + n ν ⋅ z − 2 f ′ ( 0 ) − 2 λ ∗ ∣ z ∣ − λ ∗ ∣ x ∣ , H_\mu(-n,y_n)\ge\mu(B(\nu,n^{-1/2}))e^{2f'(0)t+\lambda_*\sqrt{t+n}\,\nu\cdot z-2f'(0)-\sqrt{2}\lambda_*|z|-\lambda_*|x|}, H μ ( − n , y n ) ≥ μ ( B ( ν , n − 1/2 )) e 2 f ′ ( 0 ) t + λ ∗ t + n ν ⋅ z − 2 f ′ ( 0 ) − 2 λ ∗ ∣ z ∣ − λ ∗ ∣ x ∣ , which, with (64) and ( t + n ) / n → 1 (t+n)/n\to1 ( t + n ) / n → 1 (65) .
Set q n ( y ) = min { 1 , 1 / ( a H μ ( − n , y ) ) } q_n(y)=\min\{1,1/(aH_\mu(-n,y))\} q n ( y ) = min { 1 , 1/ ( a H μ ( − n , y ))} min { a H μ ( − n , y ) , 1 } = a H μ ( − n , y ) q n ( y ) \min\{aH_\mu(-n,y),1\}=aH_\mu(-n,y)q_n(y) min { a H μ ( − n , y ) , 1 } = a H μ ( − n , y ) q n ( y )
( L t + n [ min { a H μ ( − n , ⋅ ) , 1 } ] ) ( x ) = a e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ ν ⋅ x × ( ∫ R N ( 4 π ) − N / 2 e − ∣ z ∣ 2 / 4 q n ( y n ) d z ) d μ ( ν ) . \begin{aligned}
(L_{t+n}[\min\{aH_\mu(-n,\cdot),1\}])(x)
&=ae^{2f'(0)t}\int_{S^{N-1}}e^{\lambda_*\nu\cdot x}\\
&\qquad\times\left(\int_{\mathbb{R}^{N}}(4\pi)^{-N/2}e^{-|z|^2/4}q_n(y_n)\,dz\right)d\mu(\nu).
\end{aligned} ( L t + n [ min { a H μ ( − n , ⋅ ) , 1 }]) ( x ) = a e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ ν ⋅ x × ( ∫ R N ( 4 π ) − N /2 e − ∣ z ∣ 2 /4 q n ( y n ) d z ) d μ ( ν ) . By (65) , q n ( y n ) q_n(y_n) q n ( y n ) 1 1 1 ν ⋅ z < 0 \nu\cdot z<0 ν ⋅ z < 0 0 0 0 ν ⋅ z > 0 \nu\cdot z>0 ν ⋅ z > 0 ν ⋅ z = 0 \nu\cdot z=0 ν ⋅ z = 0 0 ≤ q n ≤ 1 0\le q_n\le1 0 ≤ q n ≤ 1 μ \mu μ ν \nu ν
I n ( ν ) : = ∫ R N ( 4 π ) − N / 2 e − ∣ z ∣ 2 / 4 q n ( y n ) d z ⟶ ∫ { z : ν ⋅ z < 0 } ( 4 π ) − N / 2 e − ∣ z ∣ 2 / 4 d z = 1 2 . \begin{aligned}
I_n(\nu)&:=\int_{\mathbb{R}^{N}}(4\pi)^{-N/2}e^{-|z|^2/4}q_n(y_n)\,dz\\
&\longrightarrow\int_{\{z:\nu\cdot z<0\}}(4\pi)^{-N/2}e^{-|z|^2/4}\,dz=\frac{1}{2}.
\end{aligned} I n ( ν ) := ∫ R N ( 4 π ) − N /2 e − ∣ z ∣ 2 /4 q n ( y n ) d z ⟶ ∫ { z : ν ⋅ z < 0 } ( 4 π ) − N /2 e − ∣ z ∣ 2 /4 d z = 2 1 . Moreover,
0 ≤ e λ ∗ ν ⋅ x I n ( ν ) ≤ e λ ∗ ∣ x ∣ , μ ( S N − 1 ) < ∞ . 0\le e^{\lambda_*\nu\cdot x}I_n(\nu)\le e^{\lambda_*|x|},\qquad\mu(S^{N-1})<\infty. 0 ≤ e λ ∗ ν ⋅ x I n ( ν ) ≤ e λ ∗ ∣ x ∣ , μ ( S N − 1 ) < ∞. Applying dominated convergence to the outer integral therefore yields
( L t + n [ min { a H μ ( − n , ⋅ ) , 1 } ] ) ( x ) ⟶ a 2 e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ ν ⋅ x d μ ( ν ) = a 2 H μ ( t , x ) . (L_{t+n}[\min\{aH_\mu(-n,\cdot),1\}])(x)\longrightarrow\frac{a}{2}e^{2f'(0)t}\int_{S^{N-1}}e^{\lambda_*\nu\cdot x}\,d\mu(\nu)=\frac{a}{2}H_\mu(t,x). ( L t + n [ min { a H μ ( − n , ⋅ ) , 1 }]) ( x ) ⟶ 2 a e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ ν ⋅ x d μ ( ν ) = 2 a H μ ( t , x ) . as n → + ∞ n\to+\infty n → + ∞
The preceding Gaussian estimate rules out a nonzero nonlinear entire solution bounded above by a critical exponential field.
Proposition 3.2. If an entire solution 0 ≤ V ≤ 1 0 \le V \le1 0 ≤ V ≤ 1 (1) satisfies V ( t , x ) ≤ H μ ( t , x ) V(t,x) \le H_\mu(t,x) V ( t , x ) ≤ H μ ( t , x ) H μ H_\mu H μ (63) , then V ≡ 0 V \equiv0 V ≡ 0
Proof. More generally, suppose V ≤ a H μ V \le aH_\mu V ≤ a H μ a > 0 a > 0 a > 0 ∂ t H μ − Δ H μ = f ′ ( 0 ) H μ \partial_t H_\mu-\Delta H_\mu=f'(0)H_\mu ∂ t H μ − Δ H μ = f ′ ( 0 ) H μ a H μ aH_\mu a H μ (1) . For fixed ( t , x ) (t,x) ( t , x ) n > max { 0 , − t } n>\max\{0,-t\} n > max { 0 , − t }
V ( t , x ) ≤ ( L t + n [ min { a H μ ( − n , ⋅ ) , 1 } ] ) ( x ) . V(t,x) \le\left(L_{t+n}[\min\{aH_\mu(-n,\cdot),1\}]\right)(x). V ( t , x ) ≤ ( L t + n [ min { a H μ ( − n , ⋅ ) , 1 }] ) ( x ) . Lemma 3.1 gives V ( t , x ) ≤ ( a / 2 ) H μ ( t , x ) V(t,x) \le(a/2)H_\mu(t,x) V ( t , x ) ≤ ( a /2 ) H μ ( t , x ) V ( t , x ) = 0 V(t,x)=0 V ( t , x ) = 0
Consequently, evolution from capped critical exponential data converges to zero as the initial time tends to the remote past.
Corollary 3.3. For every a > 0 a>0 a > 0
S t + T [ min { a H μ ( − T , ⋅ ) , 1 } ] → 0 in C l o c 1 , 2 ( R × R N ) as T → ∞ . (66) S_{t+T}[\min\{aH_\mu(-T,\cdot),1\}] \to0 \quad\text{in } C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^{N}) \text{ as } T\to\infty. \tag*{(66)} S t + T [ min { a H μ ( − T , ⋅ ) , 1 }] → 0 in C loc 1 , 2 ( R × R N ) as T → ∞. (66) Proof. Since a H μ aH_\mu a H μ f ′ ( 0 ) a H μ − f ( a H μ ) ≥ 0 f'(0)aH_\mu-f(aH_\mu)\ge0 f ′ ( 0 ) a H μ − f ( a H μ ) ≥ 0 { a H μ < 1 } \{aH_\mu<1\} { a H μ < 1 } min { a H μ , 1 } \min\{aH_\mu,1\} min { a H μ , 1 } S t + T [ min { a H μ ( − T , ⋅ ) , 1 } ] S_{t+T}[\min\{aH_\mu(-T,\cdot),1\}] S t + T [ min { a H μ ( − T , ⋅ ) , 1 }] (1) which is less than a H μ aH_\mu a H μ
We next recall the solutions generated by finite families of traveling fronts and establish their uniqueness between the corresponding barriers.
Write
X : = ( S N − 1 × [ c ∗ , + ∞ ) ) ∪ { ∞ } , λ ( c ) : = c − c 2 − 4 f ′ ( 0 ) 2 for c ≥ c ∗ . X := (\mathbb{S}^{N-1}\times[c_\ast,+\infty))\cup\{\infty\},\qquad\lambda(c):=\frac{c-\sqrt{c^2-4f'(0)}}{2}\quad\text{for }c\ge c_\ast. X := ( S N − 1 × [ c ∗ , + ∞ )) ∪ { ∞ } , λ ( c ) := 2 c − c 2 − 4 f ′ ( 0 ) for c ≥ c ∗ . The point ∞ \infty ∞
ϑ = ∑ i = 1 m a i δ ( ν i , c i ) + a 0 δ ∞ \vartheta=\sum_{i=1}^{m}a_i\delta_{(\nu_i,c_i)}+a_0\delta_\infty ϑ = i = 1 ∑ m a i δ ( ν i , c i ) + a 0 δ ∞ be a finite positive atomic measure on X X X m ≥ 0 m\ge0 m ≥ 0 a 0 ≥ 0 a_0\ge0 a 0 ≥ 0 a i > 0 a_i>0 a i > 0 ν i ∈ S N − 1 \nu_i\in\mathbb{S}^{N-1} ν i ∈ S N − 1 c i ≥ c ∗ c_i\ge c_\ast c i ≥ c ∗ i = 1 , … , m i=1,\ldots,m i = 1 , … , m ( ν i , c i ) (\nu_i,c_i) ( ν i , c i ) h i h_i h i a i a_i a i
v i ( t , x ) = ϕ c i ( x ⋅ ν i + c i t + h i ) , i = 1 , … , m . v_i(t,x)=\phi_{c_i}(x\cdot\nu_i+c_i t+h_i),\qquad i=1,\ldots,m. v i ( t , x ) = ϕ c i ( x ⋅ ν i + c i t + h i ) , i = 1 , … , m . If a 0 > 0 a_0>0 a 0 > 0 v 0 ( t , x ) = ξ f ( t + h 0 ) v_0(t,x)=\xi_f(t+h_0) v 0 ( t , x ) = ξ f ( t + h 0 ) h 0 h_0 h 0 a 0 a_0 a 0 i = 1 , … , m i=1,\ldots,m i = 1 , … , m a 0 > 0 a_0>0 a 0 > 0 f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 u ϑ u_\vartheta u ϑ
max i v i ≤ u ϑ ≤ min { ∑ i v i , 1 } . \max_i v_i\le u_\vartheta\le\min\left\{\sum_i v_i,1\right\}. i max v i ≤ u ϑ ≤ min { i ∑ v i , 1 } . This is the finite-front construction of [28 Theorem 1.1 and Section 2] , namely
u ϑ ( t , x ) = lim T → ∞ ( S t + T [ max i v i ( − T , ⋅ ) ] ) ( x ) . (67) u_\vartheta(t,x)=\lim_{T\to\infty}\left(S_{t+T}\left[\max_i v_i(-T,\cdot)\right]\right)(x). \tag*{(67)} u ϑ ( t , x ) = T → ∞ lim ( S t + T [ i max v i ( − T , ⋅ ) ] ) ( x ) . (67) We record the uniqueness property of this solution.
Theorem 3.4. Any entire solution U U U satisfying
max i v i ≤ U ≤ min { ∑ i v i , 1 } \max_i v_i \le U \le\min\left\{\sum_i v_i,1\right\} i max v i ≤ U ≤ min { i ∑ v i , 1 } coincides with u ϑ u_{\vartheta} u ϑ
Proof. Let U U U V V V ( t , x ) (t,x) ( t , x ) k k k v k ( t , x ) = max i v i ( t , x ) v_k(t,x)=\max_i v_i(t,x) v k ( t , x ) = max i v i ( t , x ) v k ≤ ∑ i v i v_k\le\sum_i v_i v k ≤ ∑ i v i v k ≤ 1 v_k\le1 v k ≤ 1 0 ≤ v i ≤ v k 0\le v_i\le v_k 0 ≤ v i ≤ v k v i ≤ v i v k v_i\le\sqrt{v_i v_k} v i ≤ v i v k
0 ≤ min { ∑ i v i , 1 } − max i v i ≤ ∑ i ≠ k v i ≤ ∑ i ≠ k v i v k ≤ ∑ i < j v i v j . 0\le\min\left\{\sum_i v_i,1\right\}-\max_i v_i\le\sum_{i\ne k}v_i\le\sum_{i\ne k}\sqrt{v_i v_k}\le\sum_{i<j}\sqrt{v_i v_j}. 0 ≤ min { i ∑ v i , 1 } − i max v i ≤ i = k ∑ v i ≤ i = k ∑ v i v k ≤ i < j ∑ v i v j . Estimate (61) gives, for T > − t T>-t T > − t
∣ U ( t , x ) − V ( t , x ) ∣ ≤ ∑ i < j ( L T + t [ v i ( − T , ⋅ ) v j ( − T , ⋅ ) ] ) ( x ) . \lvert U(t,x)-V(t,x)\rvert\le\sum_{i<j}\left(L_{T+t}\left[\sqrt{v_i(-T,\cdot)v_j(-T,\cdot)}\right]\right)(x). ∣ U ( t , x ) − V ( t , x )∣ ≤ i < j ∑ ( L T + t [ v i ( − T , ⋅ ) v j ( − T , ⋅ ) ] ) ( x ) . The normalized tails established in (4) –(7) , together with 0 < ϕ c < 1 0<\phi_c<1 0 < ϕ c < 1
ϕ c ( s ) ≤ C c e λ ( c ) s for c > c ∗ , ϕ ∗ ( s ) ≤ C ( 1 + ∣ s ∣ ) e λ ∗ s , for s ∈ R . \phi_c(s)\le C_c e^{\lambda(c)s}\quad\text{for }c>c_*,\qquad\phi_*(s)\le C(1+\lvert s\rvert)e^{\lambda_*s},\quad\text{for }s\in\mathbb{R}. ϕ c ( s ) ≤ C c e λ ( c ) s for c > c ∗ , ϕ ∗ ( s ) ≤ C ( 1 + ∣ s ∣) e λ ∗ s , for s ∈ R . Substituting s = y ⋅ ν i − c i T + h i s=y\cdot\nu_i-c_iT+h_i s = y ⋅ ν i − c i T + h i 1 + ∣ s ∣ ≤ 1 + ∣ y ∣ + c i T + ∣ h i ∣ 1+\lvert s\rvert\le1+\lvert y\rvert+c_iT+\lvert h_i\rvert 1 + ∣ s ∣ ≤ 1 + ∣ y ∣ + c i T + ∣ h i ∣ T ≥ 1 T\ge1 T ≥ 1 y ∈ R N y\in\mathbb{R}^N y ∈ R N
v i ( − T , y ) = ϕ c i ( y ⋅ ν i − c i T + h i ) ≤ K i ( 1 + T + ∣ y ∣ ) e λ ( c i ) y ⋅ ν i − c i λ ( c i ) T . v_i(-T,y)=\phi_{c_i}(y\cdot\nu_i-c_iT+h_i)\le K_i(1+T+\lvert y\rvert)e^{\lambda(c_i)y\cdot\nu_i-c_i\lambda(c_i)T}. v i ( − T , y ) = ϕ c i ( y ⋅ ν i − c i T + h i ) ≤ K i ( 1 + T + ∣ y ∣) e λ ( c i ) y ⋅ ν i − c i λ ( c i ) T . Here K i K_i K i e λ ( c i ) h i e^{\lambda(c_i)h_i} e λ ( c i ) h i T T T y y y (8) gives v 0 ( − T , y ) = ξ f ( − T + h 0 ) ≤ K 0 e − f ′ ( 0 ) T v_0(-T,y)=\xi_f(-T+h_0)\le K_0e^{-f'(0)T} v 0 ( − T , y ) = ξ f ( − T + h 0 ) ≤ K 0 e − f ′ ( 0 ) T c i λ ( c i ) = f ′ ( 0 ) + λ ( c i ) 2 c_i\lambda(c_i)=f'(0)+\lambda(c_i)^2 c i λ ( c i ) = f ′ ( 0 ) + λ ( c i ) 2 L t L_t L t 1 ≤ i < j ≤ m 1\le i<j\le m 1 ≤ i < j ≤ m R > 0 R>0 R > 0 T T T
sup ( t , x ) ∈ [ − R , R ] × B R ‾ ( L T + t [ v i ( − T , ⋅ ) v j ( − T , ⋅ ) ] ) ( x ) ≤ K i , j , R ( 1 + T ) e − ∣ λ ( c i ) ν i − λ ( c j ) ν j ∣ 2 T / 4 . \sup_{(t,x)\in[-R,R]\times\overline{B_R}}\left(L_{T+t}\left[\sqrt{v_i(-T,\cdot)v_j(-T,\cdot)}\right]\right)(x)\le K_{i,j,R}(1+T)e^{-\lvert\lambda(c_i)\nu_i-\lambda(c_j)\nu_j\rvert^2T/4}. ( t , x ) ∈ [ − R , R ] × B R sup ( L T + t [ v i ( − T , ⋅ ) v j ( − T , ⋅ ) ] ) ( x ) ≤ K i , j , R ( 1 + T ) e − ∣ λ ( c i ) ν i − λ ( c j ) ν j ∣ 2 T /4 . For a pair involving the homogeneous solution, the same computation gives
sup ( t , x ) ∈ [ − R , R ] × B R ‾ ( L T + t [ v 0 ( − T , ⋅ ) v i ( − T , ⋅ ) ] ) ( x ) ≤ K 0 , i , R ( 1 + T ) e − λ ( c i ) 2 T / 4 . \sup_{(t,x)\in[-R,R]\times\overline{B_R}}\left(L_{T+t}\left[\sqrt{v_0(-T,\cdot)v_i(-T,\cdot)}\right]\right)(x)\le K_{0,i,R}(1+T)e^{-\lambda(c_i)^2T/4}. ( t , x ) ∈ [ − R , R ] × B R sup ( L T + t [ v 0 ( − T , ⋅ ) v i ( − T , ⋅ ) ] ) ( x ) ≤ K 0 , i , R ( 1 + T ) e − λ ( c i ) 2 T /4 . Since λ ( c ) \lambda(c) λ ( c ) c ≥ c ∗ c\ge c_* c ≥ c ∗ ( ν i , c i ) (\nu_i,c_i) ( ν i , c i ) λ ( c i ) ν i \lambda(c_i)\nu_i λ ( c i ) ν i T → ∞ T\to\infty T → ∞ U = V U=V U = V
We now prove approximation by solutions associated with finite atomic measures under a finiteness condition on the linear evolution from the remote past.
Theorem 3.5. Let U U U (1) with 0 < U < 1 0<U<1 0 < U < 1 R × R N \mathbb{R}\times\mathbb{R}^N R × R N
sup T > 0 ( L T [ U ( − T , ⋅ ) ] ) ( 0 ) < ∞ . (68) \sup_{T>0}\left(L_T[U(-T,\cdot)]\right)(0)<\infty. \tag*{(68)} T > 0 sup ( L T [ U ( − T , ⋅ )] ) ( 0 ) < ∞. (68) Then there exists a sequence of finite positive measures on X X X
ϑ k = ∑ i = 1 m k a k , i δ ( ν k , i , c k , i ) + a k , 0 δ ∞ , (69) \vartheta_k=\sum_{i=1}^{m_k}a_{k,i}\delta_{(\nu_{k,i},c_{k,i})}+a_{k,0}\delta_\infty, \tag*{(69)} ϑ k = i = 1 ∑ m k a k , i δ ( ν k , i , c k , i ) + a k , 0 δ ∞ , (69) where m k ≥ 0 m_k\ge0 m k ≥ 0 a k , 0 ≥ 0 a_{k,0}\ge0 a k , 0 ≥ 0
a k , i > 0 , ν k , i ∈ S N − 1 , c k , i ≥ c ∗ for i = 1 , … , m k , a_{k,i}>0,\qquad\nu_{k,i}\in S^{N-1},\qquad c_{k,i}\ge c_*\quad\text{for }i=1,\ldots,m_k, a k , i > 0 , ν k , i ∈ S N − 1 , c k , i ≥ c ∗ for i = 1 , … , m k , with the pairs ( ν k , i , c k , i ) (\nu_{k,i},c_{k,i}) ( ν k , i , c k , i ) k k k
u ϑ k ⟶ U in C l o c 1 , 2 ( R × R N ) as k → ∞ . u_{\vartheta_k}\longrightarrow U\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N)\quad\text{as }k\to\infty. u ϑ k ⟶ U in C loc 1 , 2 ( R × R N ) as k → ∞. Proof. We give the construction and each limiting step.
Step 1: the associated linear solution and exact reconstruction. Put H T ( t , ⋅ ) = L t + T [ U ( − T , ⋅ ) ] H_T(t,\cdot)=L_{t+T}[U(-T,\cdot)] H T ( t , ⋅ ) = L t + T [ U ( − T , ⋅ )]
∂ T H T = L t + T [ ( f ′ ( 0 ) U − f ( U ) ) ( − T , ⋅ ) ] ≥ 0 , H T − U = ∫ − T t L t − s [ ( f ′ ( 0 ) U − f ( U ) ) ( s , ⋅ ) ] d s . \partial_T H_T=L_{t+T}[(f'(0)U-f(U))(-T,\cdot)]\ge0,\qquad H_T-U=\int_{-T}^{t}L_{t-s}[(f'(0)U-f(U))(s,\cdot)]\,\mathrm{d}s. ∂ T H T = L t + T [( f ′ ( 0 ) U − f ( U )) ( − T , ⋅ )] ≥ 0 , H T − U = ∫ − T t L t − s [( f ′ ( 0 ) U − f ( U )) ( s , ⋅ )] d s . To obtain a local bound for H T H_T H T Theorem 2.1 first gives U ( − T , y + x ) ≤ e λ ∗ ∣ x ∣ U ( − T , y ) U(-T,y+x)\le e^{\lambda_*|x|}U(-T,y) U ( − T , y + x ) ≤ e λ ∗ ∣ x ∣ U ( − T , y )
H T ( t , x ) = ( L t + T [ U ( − T , ⋅ + x ) ] ) ( 0 ) ≤ e λ ∗ ∣ x ∣ ( L t + T [ U ( − T , ⋅ ) ] ) ( 0 ) = e λ ∗ ∣ x ∣ H T ( t , 0 ) . (70) H_T(t,x)=(L_{t+T}[U(-T,\cdot+x)])(0)\le e^{\lambda_*|x|}(L_{t+T}[U(-T,\cdot)])(0)=e^{\lambda_*|x|}H_T(t,0). \tag*{(70)} H T ( t , x ) = ( L t + T [ U ( − T , ⋅ + x )]) ( 0 ) ≤ e λ ∗ ∣ x ∣ ( L t + T [ U ( − T , ⋅ )]) ( 0 ) = e λ ∗ ∣ x ∣ H T ( t , 0 ) . (70) Next, the estimate 0 < ∂ t ln U ≤ 2 f ′ ( 0 ) 0<\partial_t\ln U\le2f'(0) 0 < ∂ t ln U ≤ 2 f ′ ( 0 )
U ( − T , y ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ U ( − T − t , y ) . U(-T,y)\le e^{2f'(0)|t|}U(-T-t,y). U ( − T , y ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ U ( − T − t , y ) . Applying the same positive operator L T + t L_{T+t} L T + t
H T ( t , 0 ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ ( L T + t [ U ( − T − t , ⋅ ) ] ) ( 0 ) = e 2 f ′ ( 0 ) ∣ t ∣ H T + t ( 0 , 0 ) . H_T(t,0)\le e^{2f'(0)|t|}(L_{T+t}[U(-T-t,\cdot)])(0)=e^{2f'(0)|t|}H_{T+t}(0,0). H T ( t , 0 ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ ( L T + t [ U ( − T − t , ⋅ )]) ( 0 ) = e 2 f ′ ( 0 ) ∣ t ∣ H T + t ( 0 , 0 ) . Thus, for every R > 0 R>0 R > 0 T > R T>R T > R
sup ( t , x ) ∈ [ − R , R ] × B R ‾ H T ( t , x ) ≤ e ( λ ∗ + 2 f ′ ( 0 ) ) R sup S > 0 ( L S [ U ( − S , ⋅ ) ] ) ( 0 ) < ∞ \sup_{(t,x)\in[-R,R]\times\overline{B_R}}H_T(t,x)\le e^{(\lambda_*+2f'(0))R}\sup_{S>0}(L_S[U(-S,\cdot)])(0)<\infty ( t , x ) ∈ [ − R , R ] × B R sup H T ( t , x ) ≤ e ( λ ∗ + 2 f ′ ( 0 )) R S > 0 sup ( L S [ U ( − S , ⋅ )]) ( 0 ) < ∞ by (68) . This bound is independent of T T T H T → H H_T\to H H T → H C l o c ∞ ( R × R N ) C^\infty_{\mathrm{loc}}(\mathbb{R}\times\mathbb{R}^N) C loc ∞ ( R × R N ) H H H (17) to the positive heat solution e − f ′ ( 0 ) t H ( t , x ) e^{-f'(0)t}H(t,x) e − f ′ ( 0 ) t H ( t , x ) λ ∗ \lambda_* λ ∗ μ \mu μ R N \mathbb{R}^N R N
H ( t , x ) = ∫ R N e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) , μ ( R N ) = H ( 0 , 0 ) < ∞ . H(t,x)=\int_{\mathbb{R}^N}e^{\lambda_*x\cdot\xi+f'(0)(1+|\xi|^2)t}\,\mathrm{d}\mu(\xi),\qquad\mu(\mathbb{R}^N)=H(0,0)<\infty. H ( t , x ) = ∫ R N e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) , μ ( R N ) = H ( 0 , 0 ) < ∞. We next show that μ \mu μ B 1 ‾ \overline{B_1} B 1 (70) gives H ( 0 , s e ) ≤ e λ ∗ s H ( 0 , 0 ) H(0,se)\le e^{\lambda_*s}H(0,0) H ( 0 , se ) ≤ e λ ∗ s H ( 0 , 0 ) s ≥ 0 s\ge0 s ≥ 0 e e e ε > 0 \varepsilon>0 ε > 0
e λ ∗ s ( 1 + ε ) μ ( { ξ : e ⋅ ξ ≥ 1 + ε } ) ≤ H ( 0 , s e ) ≤ e λ ∗ s H ( 0 , 0 ) . e^{\lambda_*s(1+\varepsilon)}\mu(\{\xi:e\cdot\xi\ge1+\varepsilon\})\le H(0,se)\le e^{\lambda_*s}H(0,0). e λ ∗ s ( 1 + ε ) μ ({ ξ : e ⋅ ξ ≥ 1 + ε }) ≤ H ( 0 , se ) ≤ e λ ∗ s H ( 0 , 0 ) . Dividing by e λ ∗ s ( 1 + ε ) e^{\lambda_*s(1+\varepsilon)} e λ ∗ s ( 1 + ε ) s → ∞ s\to\infty s → ∞ μ ( { ξ : e ⋅ ξ ≥ 1 + ε } ) = 0 \mu(\{\xi:e\cdot\xi\ge1+\varepsilon\})=0 μ ({ ξ : e ⋅ ξ ≥ 1 + ε }) = 0 e e e ε \varepsilon ε μ ( R N ∖ B 1 ‾ ) = 0 \mu(\mathbb{R}^N\setminus\overline{B_1})=0 μ ( R N ∖ B 1 ) = 0
H ( t , x ) = ∫ B 1 ‾ e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) , μ ( B 1 ‾ ) = H ( 0 , 0 ) < ∞ . H(t,x)=\int_{\overline{B_1}}e^{\lambda_*x\cdot\xi+f'(0)(1+|\xi|^2)t}\,\mathrm{d}\mu(\xi),\qquad\mu(\overline{B_1})=H(0,0)<\infty. H ( t , x ) = ∫ B 1 e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) , μ ( B 1 ) = H ( 0 , 0 ) < ∞. In particular, L t + T [ H ( − T , ⋅ ) ] = H ( t , ⋅ ) L_{t+T}[H(-T,\cdot)]=H(t,\cdot) L t + T [ H ( − T , ⋅ )] = H ( t , ⋅ ) T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t } H t − Δ H − f ( H ) = F 1 ( H ) ≥ 0 H_t-\Delta H-f(H)=F_1(H)\ge0 H t − Δ H − f ( H ) = F 1 ( H ) ≥ 0 { H < 1 } \{H<1\} { H < 1 } f ( 1 ) = 0 f(1)=0 f ( 1 ) = 0 min { H , 1 } \min\{H,1\} min { H , 1 }
V H ( t , ⋅ ) : = lim T → ∞ S t + T [ min { H ( − T , ⋅ ) , 1 } ] , V_H(t,\cdot):=\lim_{T\to\infty}S_{t+T}[\min\{H(-T,\cdot),1\}], V H ( t , ⋅ ) := T → ∞ lim S t + T [ min { H ( − T , ⋅ ) , 1 }] , with convergence in C l o c 1 ( R × R N ) C^1_{\mathrm{loc}}(\mathbb{R}\times\mathbb{R}^N) C loc 1 ( R × R N ) U ≤ H U\le H U ≤ H U ≤ 1 U\le1 U ≤ 1 U ( − T , ⋅ ) ≤ min { H ( − T , ⋅ ) , 1 } U(-T,\cdot)\le\min\{H(-T,\cdot),1\} U ( − T , ⋅ ) ≤ min { H ( − T , ⋅ ) , 1 } T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t } (61) therefore give
0 ≤ S t + T [ min { H ( − T , ⋅ ) , 1 } ] − U ( t , ⋅ ) ≤ L t + T [ H ( − T , ⋅ ) − U ( − T , ⋅ ) ] = H ( t , ⋅ ) − H T ( t , ⋅ ) . 0\le S_{t+T}[\min\{H(-T,\cdot),1\}]-U(t,\cdot)\le L_{t+T}[H(-T,\cdot)-U(-T,\cdot)]=H(t,\cdot)-H_T(t,\cdot). 0 ≤ S t + T [ min { H ( − T , ⋅ ) , 1 }] − U ( t , ⋅ ) ≤ L t + T [ H ( − T , ⋅ ) − U ( − T , ⋅ )] = H ( t , ⋅ ) − H T ( t , ⋅ ) . Letting T → ∞ T \to\infty T → ∞ H T → H H_T \to H H T → H V H = U V_H = U V H = U
Step 2: reconstruction inside the critical sphere and exclusion of boundary mass. For a finite positive Borel measure η \eta η B ‾ r \overline{B}_r B r 0 ≤ r < 1 0 \le r < 1 0 ≤ r < 1
h ( t , x ) : = ∫ B ‾ r e λ ∗ ⋅ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d η ( ξ ) , ( t , x ) ∈ R × R N . (71) h(t,x) := \int_{\overline{B}_r} e^{\lambda^*\cdot x\cdot\xi+f'(0)(1+|\xi|^2)t}\,d\eta(\xi), \qquad(t,x) \in\mathbb{R} \times\mathbb{R}^N. \tag*{(71)} h ( t , x ) := ∫ B r e λ ∗ ⋅ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d η ( ξ ) , ( t , x ) ∈ R × R N . (71) Choose
1 < p ≤ 2 , p r 2 < 1 , c = f ′ ( 0 ) ( p − 1 ) ( 1 − p r 2 ) > 0 , A ≥ max { 1 , M / c } . (72) 1 < p \le2,\qquad pr^2 < 1,\qquad c=f'(0)(p-1)(1-pr^2)>0,\qquad A\ge\max\{1,M/c\}. \tag*{(72)} 1 < p ≤ 2 , p r 2 < 1 , c = f ′ ( 0 ) ( p − 1 ) ( 1 − p r 2 ) > 0 , A ≥ max { 1 , M / c } . (72) Then ( ∂ t − Δ − f ′ ( 0 ) ) h p ≥ c h p (\partial_t-\Delta-f'(0))h^p\ge ch^p ( ∂ t − Δ − f ′ ( 0 )) h p ≥ c h p T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t }
L t + T [ h ( − T , ⋅ ) p ] ≤ e − c ( t + T ) h ( t , ⋅ ) p . (73) L_{t+T}[h(-T,\cdot)^p]\le e^{-c(t+T)}h(t,\cdot)^p. \tag*{(73)} L t + T [ h ( − T , ⋅ ) p ] ≤ e − c ( t + T ) h ( t , ⋅ ) p . (73) To prove (73) , the case η = 0 \eta=0 η = 0 h > 0 h>0 h > 0 ( t , x ) (t,x) ( t , x ) T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t }
d π t , x ( ξ ) : = e λ ∗ ⋅ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t h ( t , x ) d η ( ξ ) , π t , x ( B ‾ r ) = 1. d\pi_{t,x}(\xi):=\frac{e^{\lambda^*\cdot x\cdot\xi+f'(0)(1+|\xi|^2)t}}{h(t,x)}\,d\eta(\xi),\qquad\pi_{t,x}(\overline{B}_r)=1. d π t , x ( ξ ) := h ( t , x ) e λ ∗ ⋅ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d η ( ξ ) , π t , x ( B r ) = 1. Since p > 1 p>1 p > 1 y ∈ R N y\in\mathbb{R}^N y ∈ R N
( h ( − T , y ) h ( t , x ) ) p = ( ∫ B ‾ r e λ ∗ ⋅ ξ ⋅ ( y − x ) − f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) ( t + T ) d π t , x ( ξ ) ) p ≤ ∫ B ‾ r e p λ ∗ ⋅ ξ ⋅ ( y − x ) − p f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) ( t + T ) d π t , x ( ξ ) . \left(\frac{h(-T,y)}{h(t,x)}\right)^p
=\left(\int_{\overline{B}_r}e^{\lambda^*\cdot\xi\cdot(y-x)-f'(0)(1+|\xi|^2)(t+T)}\,d\pi_{t,x}(\xi)\right)^p
\le\int_{\overline{B}_r}e^{p\lambda^*\cdot\xi\cdot(y-x)-pf'(0)(1+|\xi|^2)(t+T)}\,d\pi_{t,x}(\xi). ( h ( t , x ) h ( − T , y ) ) p = ( ∫ B r e λ ∗ ⋅ ξ ⋅ ( y − x ) − f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) ( t + T ) d π t , x ( ξ ) ) p ≤ ∫ B r e p λ ∗ ⋅ ξ ⋅ ( y − x ) − p f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) ( t + T ) d π t , x ( ξ ) . Notice that
1 ( 4 π ( t + T ) ) N / 2 ∫ R N e − ∣ y − x ∣ 2 4 ( t + T ) + p λ ∗ ⋅ ξ ⋅ ( y − x ) d y = e p 2 f ′ ( 0 ) ∣ ξ ∣ 2 ( t + T ) . \frac{1}{(4\pi(t+T))^{N/2}}\int_{\mathbb{R}^N}e^{-\frac{|y-x|^2}{4(t+T)}+p\lambda^*\cdot\xi\cdot(y-x)}\,dy=e^{p^2f'(0)|\xi|^2(t+T)}. ( 4 π ( t + T ) ) N /2 1 ∫ R N e − 4 ( t + T ) ∣ y − x ∣ 2 + p λ ∗ ⋅ ξ ⋅ ( y − x ) d y = e p 2 f ′ ( 0 ) ∣ ξ ∣ 2 ( t + T ) . Thus, using L t + T = e f ′ ( 0 ) ( t + T ) P t + T L_{t+T}=e^{f'(0)(t+T)}P_{t+T} L t + T = e f ′ ( 0 ) ( t + T ) P t + T
( L t + T [ h ( − T , ⋅ ) p ] ) ( x ) h ( t , x ) p ≤ ∫ B ‾ r e f ′ ( 0 ) [ 1 − p ( 1 + ∣ ξ ∣ 2 ) + p 2 ∣ ξ ∣ 2 ] ( t + T ) d π t , x ( ξ ) = ∫ B ‾ r e − f ′ ( 0 ) ( p − 1 ) ( 1 − p ∣ ξ ∣ 2 ) ( t + T ) d π t , x ( ξ ) ≤ e − c ( t + T ) . \begin{aligned}
\frac{(L_{t+T}[h(-T,\cdot)^p])(x)}{h(t,x)^p}
\le\int_{\overline{B}_r}e^{f'(0)[1-p(1+|\xi|^2)+p^2|\xi|^2](t+T)}\,d\pi_{t,x}(\xi) \\
&=\int_{\overline{B}_r}e^{-f'(0)(p-1)(1-p|\xi|^2)(t+T)}\,d\pi_{t,x}(\xi) \\
&\le e^{-c(t+T)}.
\end{aligned} h ( t , x ) p ( L t + T [ h ( − T , ⋅ ) p ]) ( x ) ≤ ∫ B r e f ′ ( 0 ) [ 1 − p ( 1 + ∣ ξ ∣ 2 ) + p 2 ∣ ξ ∣ 2 ] ( t + T ) d π t , x ( ξ ) = ∫ B r e − f ′ ( 0 ) ( p − 1 ) ( 1 − p ∣ ξ ∣ 2 ) ( t + T ) d π t , x ( ξ ) ≤ e − c ( t + T ) . This proves (73) . Define
ω : = max { h − A h p , 0 } . \omega:=\max\{h-Ah^p,0\}. ω := max { h − A h p , 0 } . On { ω > 0 } \{\omega>0\} { ω > 0 } ω = h − A h p \omega=h-Ah^p ω = h − A h p A h p − 1 < 1 Ah^{p-1}<1 A h p − 1 < 1 0 < ω ≤ h < 1 0<\omega\le h<1 0 < ω ≤ h < 1 1 < p ≤ 2 1<p\le2 1 < p ≤ 2 A c ≥ M Ac\ge M A c ≥ M
ω t − Δ ω − f ( ω ) = − A ( ∂ t − Δ − f ′ ( 0 ) ) h p + F 1 ( ω ) ≤ − A c h p + M h 2 ≤ ( M − A c ) h p ≤ 0. \omega_t-\Delta\omega-f(\omega)=-A(\partial_t-\Delta-f'(0))h^p+F_1(\omega)\le-Ach^p+Mh^2\le(M-Ac)h^p\le0. ω t − Δ ω − f ( ω ) = − A ( ∂ t − Δ − f ′ ( 0 )) h p + F 1 ( ω ) ≤ − A c h p + M h 2 ≤ ( M − A c ) h p ≤ 0. Also h t − Δ h − f ( h ) = F 1 ( h ) ≥ 0 h_t-\Delta h-f(h)=F_1(h)\ge0 h t − Δ h − f ( h ) = F 1 ( h ) ≥ 0 { h < 1 } \{h<1\} { h < 1 } f ( 0 ) = f ( 1 ) = 0 f(0)=f(1)=0 f ( 0 ) = f ( 1 ) = 0 ω \omega ω min { h , 1 } \min\{h,1\} min { h , 1 } T 2 > T 1 > max { 0 , − t } T_2>T_1>\max\{0,-t\} T 2 > T 1 > max { 0 , − t }
ω ( t , x ) ≤ ( S t + T 2 [ min { h ( − T 2 , ⋅ ) , 1 } ] ) ( x ) ≤ ( S t + T 1 [ min { h ( − T 1 , ⋅ ) , 1 } ] ) ( x ) ≤ min { h ( t , x ) , 1 } . \omega(t,x)\le(S_{t+T_2}[\min\{h(-T_2,\cdot),1\}])(x)
\le(S_{t+T_1}[\min\{h(-T_1,\cdot),1\}])(x)
\le\min\{h(t,x),1\}. ω ( t , x ) ≤ ( S t + T 2 [ min { h ( − T 2 , ⋅ ) , 1 }]) ( x ) ≤ ( S t + T 1 [ min { h ( − T 1 , ⋅ ) , 1 }]) ( x ) ≤ min { h ( t , x ) , 1 } . Monotonicity and parabolic estimates therefore yield an entire solution
V h ( t , ⋅ ) : = lim T → ∞ S t + T [ min { h ( − T , ⋅ ) , 1 } ] , in C l o c 1 , 2 ( R × R N ) , (74) V_h(t,\cdot):=\lim_{T\to\infty}S_{t+T}[\min\{h(-T,\cdot),1\}],\qquad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N), \tag*{(74)} V h ( t , ⋅ ) := T → ∞ lim S t + T [ min { h ( − T , ⋅ ) , 1 }] , in C loc 1 , 2 ( R × R N ) , (74) with ω ≤ V h ≤ min { h , 1 } \omega\le V_h\le\min\{h,1\} ω ≤ V h ≤ min { h , 1 }
0 ≤ h − V h ≤ h − ω = min { h , A h p } ≤ A h p . 0\le h-V_h\le h-\omega=\min\{h,Ah^p\}\le Ah^p. 0 ≤ h − V h ≤ h − ω = min { h , A h p } ≤ A h p . For uniqueness, let W 1 , W 2 ∈ E W_1,W_2\in\mathcal{E} W 1 , W 2 ∈ E 0 ≤ h − W i ≤ A i h p 0\le h-W_i\le A_i h^p 0 ≤ h − W i ≤ A i h p A i > 0 A_i>0 A i > 0 T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t } (61) and (73) give
∣ W 1 ( t , x ) − W 2 ( t , x ) ∣ ≤ ( L t + T [ ∣ W 1 ( − T , ⋅ ) − W 2 ( − T , ⋅ ) ∣ ] ) ( x ) ≤ ( A 1 + A 2 ) ( L t + T [ h ( − T , ⋅ ) p ] ) ( x ) ≤ ( A 1 + A 2 ) e − c ( t + T ) h ( t , x ) p → 0 , as T → ∞ . \begin{aligned}
|W_1(t,x)-W_2(t,x)|&\le\left(L_{t+T}[|W_1(-T,\cdot)-W_2(-T,\cdot)|]\right)(x)\\
&\le(A_1+A_2)\left(L_{t+T}[h(-T,\cdot)^p]\right)(x)\\
&\le(A_1+A_2)e^{-c(t+T)}h(t,x)^p\to0,\qquad\text{as }T\to\infty.
\end{aligned} ∣ W 1 ( t , x ) − W 2 ( t , x ) ∣ ≤ ( L t + T [ ∣ W 1 ( − T , ⋅ ) − W 2 ( − T , ⋅ ) ∣ ] ) ( x ) ≤ ( A 1 + A 2 ) ( L t + T [ h ( − T , ⋅ ) p ] ) ( x ) ≤ ( A 1 + A 2 ) e − c ( t + T ) h ( t , x ) p → 0 , as T → ∞. Hence W 1 = W 2 W_1=W_2 W 1 = W 2
For continuity, let η k ⇀ η \eta_k\rightharpoonup\eta η k ⇀ η B ‾ r \overline{B}_r B r h k , h h_k,h h k , h (71) . Compactness of B ‾ r \overline{B}_r B r
h k → h in C l o c ∞ ( R × R N ) . h_k\to h\quad\text{in }C_{\mathrm{loc}}^\infty(\mathbb{R}\times\mathbb{R}^N). h k → h in C loc ∞ ( R × R N ) . The constants p p p c c c A A A k k k
0 ≤ V h k ≤ 1 , 0 ≤ h k − V h k ≤ A h k p . 0\le V_{h_k}\le1,\qquad0\le h_k-V_{h_k}\le Ah_k^p. 0 ≤ V h k ≤ 1 , 0 ≤ h k − V h k ≤ A h k p . Parabolic estimates make ( V h k ) (V_{h_k}) ( V h k ) C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) W ∈ E W\in\mathcal{E} W ∈ E 0 ≤ h − W ≤ A h p 0\le h-W\le Ah^p 0 ≤ h − W ≤ A h p W = V h W=V_h W = V h
V h k → V h in C l o c 1 , 2 ( R × R N ) . V_{h_k}\to V_h\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N). V h k → V h in C loc 1 , 2 ( R × R N ) . If η = 0 \eta=0 η = 0 h = 0 h=0 h = 0 W = V h = 0 W=V_h=0 W = V h = 0
Split H = H i n t + H b d H=H_{\mathrm{int}}+H_{\mathrm{bd}} H = H int + H bd ∣ ξ ∣ < 1 |\xi|<1 ∣ ξ ∣ < 1 ∣ ξ ∣ = 1 |\xi|=1 ∣ ξ ∣ = 1
H b d ( t , x ) = ∫ ∂ B 1 e λ ∗ x ⋅ ξ + 2 f ′ ( 0 ) t d μ ( ξ ) . H_{\mathrm{bd}}(t,x)=\int_{\partial B_1}e^{\lambda^*\!x\cdot\xi+2f'(0)t}\,d\mu(\xi). H bd ( t , x ) = ∫ ∂ B 1 e λ ∗ x ⋅ ξ + 2 f ′ ( 0 ) t d μ ( ξ ) . Since μ ( ∂ B 1 ) < ∞ \mu(\partial B_1)<\infty μ ( ∂ B 1 ) < ∞ Proposition 3.2 and (66) give
V H b d ( t , ⋅ ) = lim T → ∞ S t + T [ min { H b d ( − T , ⋅ ) , 1 } ] = 0 , V_{H_{\mathrm{bd}}}(t,\cdot)=\lim_{T\to\infty}S_{t+T}[\min\{H_{\mathrm{bd}}(-T,\cdot),1\}]=0, V H bd ( t , ⋅ ) = T → ∞ lim S t + T [ min { H bd ( − T , ⋅ ) , 1 }] = 0 , with convergence in C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) Lemma 2.2 now give V H = V H i n t V_H=V_{H_{\mathrm{int}}} V H = V H int U ( − T , ⋅ ) ≤ H i n t ( − T , ⋅ ) U(-T,\cdot)\le H_{\mathrm{int}}(-T,\cdot) U ( − T , ⋅ ) ≤ H int ( − T , ⋅ ) L t + T L_{t+T} L t + T T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t }
H T ( t , ⋅ ) = L t + T [ U ( − T , ⋅ ) ] ≤ L t + T [ H i n t ( − T , ⋅ ) ] = H i n t ( t , ⋅ ) . H_T(t,\cdot)=L_{t+T}[U(-T,\cdot)]\le L_{t+T}[H_{\mathrm{int}}(-T,\cdot)]=H_{\mathrm{int}}(t,\cdot). H T ( t , ⋅ ) = L t + T [ U ( − T , ⋅ )] ≤ L t + T [ H int ( − T , ⋅ )] = H int ( t , ⋅ ) . Letting T → ∞ T\to\infty T → ∞ H ≤ H i n t H\le H_{\mathrm{int}} H ≤ H int H b d = 0 H_{\mathrm{bd}}=0 H bd = 0 μ ( ∂ B 1 ) = 0 \mu(\partial B_1)=0 μ ( ∂ B 1 ) = 0
Step 3: finite atomic approximation and identification with HN solutions. For 0 < r < 1 0<r<1 0 < r < 1
H r ( t , x ) : = ∫ B ‾ r e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) . H_r(t,x):=\int_{\overline{B}_r}e^{\lambda^*\!x\cdot\xi+f'(0)(1+|\xi|^2)t}\,d\mu(\xi). H r ( t , x ) := ∫ B r e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) . Since H r ≤ H H_r\le H H r ≤ H V H r ≤ V H = U V_{H_r}\le V_H=U V H r ≤ V H = U r r r p r > 1 p_r>1 p r > 1 A r > 0 A_r>0 A r > 0 c r > 0 c_r>0 c r > 0 0 ≤ H r − V H r ≤ A r H r p r 0\le H_r-V_{H_r}\le A_rH_r^{p_r} 0 ≤ H r − V H r ≤ A r H r p r (73) holds with these constants. Combining this bound with U ≤ H U \le H U ≤ H T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t }
0 ≤ U ( − T , ⋅ ) − V H r ( − T , ⋅ ) ≤ H ( − T , ⋅ ) − V H r ( − T , ⋅ ) = H ( − T , ⋅ ) − H r ( − T , ⋅ ) + H r ( − T , ⋅ ) − V H r ( − T , ⋅ ) ≤ H ( − T , ⋅ ) − H r ( − T , ⋅ ) + A r H r ( − T , ⋅ ) p r . 0 \le U(-T,\cdot)-V_{H_r}(-T,\cdot)\le H(-T,\cdot)-V_{H_r}(-T,\cdot)
=H(-T,\cdot)-H_r(-T,\cdot)+H_r(-T,\cdot)-V_{H_r}(-T,\cdot)
\le H(-T,\cdot)-H_r(-T,\cdot)+A_rH_r(-T,\cdot)^{p_r}. 0 ≤ U ( − T , ⋅ ) − V H r ( − T , ⋅ ) ≤ H ( − T , ⋅ ) − V H r ( − T , ⋅ ) = H ( − T , ⋅ ) − H r ( − T , ⋅ ) + H r ( − T , ⋅ ) − V H r ( − T , ⋅ ) ≤ H ( − T , ⋅ ) − H r ( − T , ⋅ ) + A r H r ( − T , ⋅ ) p r . Applying (61) from time − T -T − T t t t (73) that
0 ≤ U ( t , x ) − V H r ( t , x ) ≤ ( L t + T [ U ( − T , ⋅ ) − V H r ( − T , ⋅ ) ] ) ( x ) ≤ H ( t , x ) − H r ( t , x ) + A r e − c r ( t + T ) H r ( t , x ) p r . 0 \le U(t,x)-V_{H_r}(t,x)\le\left(L_{t+T}[U(-T,\cdot)-V_{H_r}(-T,\cdot)]\right)(x)
\le H(t,x)-H_r(t,x)+A_r e^{-c_r(t+T)}H_r(t,x)^{p_r}. 0 ≤ U ( t , x ) − V H r ( t , x ) ≤ ( L t + T [ U ( − T , ⋅ ) − V H r ( − T , ⋅ )] ) ( x ) ≤ H ( t , x ) − H r ( t , x ) + A r e − c r ( t + T ) H r ( t , x ) p r . First let T → ∞ T\to\infty T → ∞ r r r 0 ≤ U − V H r ≤ H − H r 0\le U-V_{H_r}\le H-H_r 0 ≤ U − V H r ≤ H − H r R > 0 R>0 R > 0
sup ( t , x ) ∈ [ − R , R ] × B R ‾ ∣ H ( t , x ) − H r ( t , x ) ∣ = sup ( t , x ) ∈ [ − R , R ] × B R ‾ ∫ { r < ∣ ξ ∣ ≤ 1 } e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) ≤ e ( λ ∗ + 2 f ′ ( 0 ) ) R μ ( { r < ∣ ξ ∣ ≤ 1 } ) . \sup_{(t,x)\in[-R,R]\times\overline{B_R}}|H(t,x)-H_r(t,x)|
=\sup_{(t,x)\in[-R,R]\times\overline{B_R}}\int_{\{r<|\xi|\le1\}}e^{\lambda_*x\cdot\xi+f'(0)(1+|\xi|^2)t}\,d\mu(\xi)
\le e^{(\lambda_*+2f'(0))R}\mu(\{r<|\xi|\le1\}). ( t , x ) ∈ [ − R , R ] × B R sup ∣ H ( t , x ) − H r ( t , x ) ∣ = ( t , x ) ∈ [ − R , R ] × B R sup ∫ { r < ∣ ξ ∣ ≤ 1 } e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t d μ ( ξ ) ≤ e ( λ ∗ + 2 f ′ ( 0 )) R μ ({ r < ∣ ξ ∣ ≤ 1 }) . Since μ \mu μ μ ( { r < ∣ ξ ∣ ≤ 1 } ) → μ ( ∂ B 1 ) = 0 \mu(\{r<|\xi|\le1\})\to\mu(\partial B_1)=0 μ ({ r < ∣ ξ ∣ ≤ 1 }) → μ ( ∂ B 1 ) = 0 r ↗ 1 r\nearrow1 r ↗ 1 H r → H H_r\to H H r → H R × R N \mathbb{R}\times\mathbb{R}^N R × R N V H r → U V_{H_r}\to U V H r → U C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) r ↗ 1 r\nearrow1 r ↗ 1
Fix r < 1 r<1 r < 1 μ ( B r ‾ ) > 0 \mu(\overline{B_r})>0 μ ( B r ) > 0
η n = ∑ j = 1 m n b n , j δ ξ n , j ⇀ μ ∣ B r ‾ , b n , j > 0 , ξ n , j ∈ B r ‾ , \eta_n=\sum_{j=1}^{m_n}b_{n,j}\delta_{\xi_{n,j}}\rightharpoonup\mu|_{\overline{B_r}},\qquad b_{n,j}>0,\quad\xi_{n,j}\in\overline{B_r}, η n = j = 1 ∑ m n b n , j δ ξ n , j ⇀ μ ∣ B r , b n , j > 0 , ξ n , j ∈ B r , with distinct atoms and ∑ j b n , j = μ ( B r ‾ ) \sum_j b_{n,j}=\mu(\overline{B_r}) ∑ j b n , j = μ ( B r )
h n ( t , x ) = ∑ j = 1 m n b n , j e λ ∗ x ⋅ ξ n , j + f ′ ( 0 ) ( 1 + ∣ ξ n , j ∣ 2 ) t . h_n(t,x)=\sum_{j=1}^{m_n}b_{n,j}e^{\lambda_*x\cdot\xi_{n,j}+f'(0)(1+|\xi_{n,j}|^2)t}. h n ( t , x ) = j = 1 ∑ m n b n , j e λ ∗ x ⋅ ξ n , j + f ′ ( 0 ) ( 1 + ∣ ξ n , j ∣ 2 ) t . The continuity proved above gives V h n → V H r V_{h_n}\to V_{H_r} V h n → V H r C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) n → ∞ n\to\infty n → ∞
We show that each V h n V_{h_n} V h n u ϑ u_\vartheta u ϑ (67) , for some nonzero finite atomic measure ϑ \vartheta ϑ X \mathcal{X} X b δ ξ b\delta_\xi b δ ξ b > 0 b>0 b > 0 0 < ∣ ξ ∣ < 1 0<|\xi|<1 0 < ∣ ξ ∣ < 1
ν = ξ ∣ ξ ∣ , c = λ ∗ 1 + ∣ ξ ∣ 2 ∣ ξ ∣ > c ∗ , z = x ⋅ ν + c t . \nu=\frac{\xi}{|\xi|},\qquad c=\lambda_*\frac{1+|\xi|^2}{|\xi|}>c_*,\qquad z=x\cdot\nu+ct. ν = ∣ ξ ∣ ξ , c = λ ∗ ∣ ξ ∣ 1 + ∣ ξ ∣ 2 > c ∗ , z = x ⋅ ν + c t . Then λ ( c ) = λ ∗ ∣ ξ ∣ \lambda(c)=\lambda_*|\xi| λ ( c ) = λ ∗ ∣ ξ ∣ h ( t , x ) = b e λ ( c ) z h(t,x)=be^{\lambda(c)z} h ( t , x ) = b e λ ( c ) z a ∈ R a\in\mathbb{R} a ∈ R y ∈ R N y\in\mathbb{R}^N y ∈ R N y ⋅ ν + c a = 0 y\cdot\nu+ca=0 y ⋅ ν + c a = 0
h ( − T , x + y ) = b e λ ( c ) ( x ⋅ ν + y ⋅ ν − c T ) = h ( − ( T + a ) , x ) . h(-T,x+y)=be^{\lambda(c)(x\cdot\nu+y\cdot\nu-cT)}=h(-(T+a),x). h ( − T , x + y ) = b e λ ( c ) ( x ⋅ ν + y ⋅ ν − c T ) = h ( − ( T + a ) , x ) . Using spatial translation invariance of S τ S_\tau S τ V h V_h V h
V h ( t + a , x + y ) = lim T → ∞ ( S t + a + T [ min { h ( − T , ⋅ + y ) , 1 } ] ) ( x ) = lim T → ∞ ( S t + ( T + a ) [ min { h ( − ( T + a ) , ⋅ ) , 1 } ] ) ( x ) = V h ( t , x ) . V_h(t+a,x+y)=\lim_{T\to\infty}\left(S_{t+a+T}[\min\{h(-T,\cdot+y),1\}]\right)(x)
=\lim_{T\to\infty}\left(S_{t+(T+a)}[\min\{h(-(T+a),\cdot),1\}]\right)(x)
=V_h(t,x). V h ( t + a , x + y ) = T → ∞ lim ( S t + a + T [ min { h ( − T , ⋅ + y ) , 1 }] ) ( x ) = T → ∞ lim ( S t + ( T + a ) [ min { h ( − ( T + a ) , ⋅ ) , 1 }] ) ( x ) = V h ( t , x ) . Thus V h ( t , x ) = ψ ( z ) V_h(t,x)=\psi(z) V h ( t , x ) = ψ ( z ) ψ \psi ψ 0 ≤ h − V h ≤ A h p 0\le h-V_h\le Ah^p 0 ≤ h − V h ≤ A h p
ψ ′ ′ − c ψ ′ + f ( ψ ) = 0 , ψ ( z ) = b e λ ( c ) z + O ( e p λ ( c ) z ) as z → − ∞ . \psi''-c\psi'+f(\psi)=0,\qquad\psi(z)=be^{\lambda(c)z}+O(e^{p\lambda(c)z})\quad\text{as }z\to-\infty. ψ ′′ − c ψ ′ + f ( ψ ) = 0 , ψ ( z ) = b e λ ( c ) z + O ( e p λ ( c ) z ) as z → − ∞. Thus V h V_h V h 0 < V h < 1 0<V_h<1 0 < V h < 1 Theorem 2.1 yields ψ ′ > 0 \psi'>0 ψ ′ > 0 ψ ( − ∞ ) = 0 \psi(-\infty)=0 ψ ( − ∞ ) = 0 ψ ( + ∞ ) = 1 \psi(+\infty)=1 ψ ( + ∞ ) = 1 ϕ c ( z ) ∼ e λ ( c ) z \phi_c(z)\sim e^{\lambda(c)z} ϕ c ( z ) ∼ e λ ( c ) z (4) –(7) ,
V h ( t , x ) = ϕ c ( x ⋅ ν + c t + ln b λ ( c ) ) . V_h(t,x)=\phi_c\left(x\cdot\nu+ct+\frac{\ln b}{\lambda(c)}\right). V h ( t , x ) = ϕ c ( x ⋅ ν + c t + λ ( c ) ln b ) . For an atom b δ 0 b\delta_0 b δ 0 V h V_h V h x x x
V h ( t , x ) = ξ f ( t + ln b f ′ ( 0 ) ) . V_h(t,x)=\xi_f\left(t+\frac{\ln b}{f'(0)}\right). V h ( t , x ) = ξ f ( t + f ′ ( 0 ) ln b ) . Now let m ≥ 1 m\ge1 m ≥ 1 b j > 0 b_j>0 b j > 0 ξ j ∈ B r ‾ \xi_j\in\overline{B_r} ξ j ∈ B r j = 1 , … , m j=1,\ldots,m j = 1 , … , m
h j ( t , x ) : = b j e λ ∗ ⋅ x ⋅ ξ j + f ′ ( 0 ) ( 1 + ∣ ξ j ∣ 2 ) t , h ( t , x ) : = ∑ j = 1 m h j ( t , x ) . h_j(t,x):=b_j e^{\lambda_*\cdot x\cdot\xi_j+f'(0)(1+|\xi_j|^2)t},\qquad h(t,x):=\sum_{j=1}^{m}h_j(t,x). h j ( t , x ) := b j e λ ∗ ⋅ x ⋅ ξ j + f ′ ( 0 ) ( 1 + ∣ ξ j ∣ 2 ) t , h ( t , x ) := j = 1 ∑ m h j ( t , x ) . For each j j j v j : = V h j v_j:=V_{h_j} v j := V h j h j h_j h j v j v_j v j ξ j ≠ 0 \xi_j\ne0 ξ j = 0 ξ j = 0 \xi_j=0 ξ j = 0 h j ≤ h h_j\le h h j ≤ h max 1 ≤ j ≤ m v j ≤ V h \max_{1\le j\le m}v_j\le V_h max 1 ≤ j ≤ m v j ≤ V h p p p A A A c c c (72) . Since all ξ j \xi_j ξ j B r ‾ \overline{B_r} B r h j h_j h j 0 ≤ h j − v j ≤ A h j p 0\le h_j-v_j\le Ah_j^p 0 ≤ h j − v j ≤ A h j p j = 1 , … , m j=1,\ldots,m j = 1 , … , m b j b_j b j (73) also holds for h = ∑ j h j h=\sum_j h_j h = ∑ j h j p p p c c c
0 ≤ min { h , 1 } − min { ∑ j v j , 1 } ≤ ∑ j ( h j − v j ) ≤ A ∑ j h j p ≤ A h p . 0\le\min\{h,1\}-\min\left\{\sum_jv_j,1\right\}\le\sum_j(h_j-v_j)\le A\sum_jh_j^p\le Ah^p. 0 ≤ min { h , 1 } − min { j ∑ v j , 1 } ≤ j ∑ ( h j − v j ) ≤ A j ∑ h j p ≤ A h p . Since min { ∑ j v j , 1 } \min\{\sum_jv_j,1\} min { ∑ j v j , 1 } (61) and (73) imply
( S t + T [ min { h ( − T , ⋅ ) , 1 } ] ) ( x ) ≤ min { ∑ j v j ( t , x ) , 1 } + A ( L t + T [ h ( − T , ⋅ ) p ] ) ( x ) ≤ min { ∑ j v j ( t , x ) , 1 } + A e − c ( t + T ) h ( t , x ) p . \begin{aligned}
\left(S_{t+T}[\min\{h(-T,\cdot),1\}]\right)(x)\le\min\left\{\sum_jv_j(t,x),1\right\}+A\left(L_{t+T}[h(-T,\cdot)^p]\right)(x) \\
&\le\min\left\{\sum_jv_j(t,x),1\right\}+Ae^{-c(t+T)}h(t,x)^p.
\end{aligned} ( S t + T [ min { h ( − T , ⋅ ) , 1 }] ) ( x ) ≤ min { j ∑ v j ( t , x ) , 1 } + A ( L t + T [ h ( − T , ⋅ ) p ] ) ( x ) ≤ min { j ∑ v j ( t , x ) , 1 } + A e − c ( t + T ) h ( t , x ) p . Letting T → ∞ T\to\infty T → ∞
max j v j ≤ V h ≤ min { ∑ j v j , 1 } . \max_jv_j\le V_h\le\min\left\{\sum_jv_j,1\right\}. j max v j ≤ V h ≤ min { j ∑ v j , 1 } . Let ϑ \vartheta ϑ X X X v 1 , … , v m v_1,\ldots,v_m v 1 , … , v m u ϑ u_\vartheta u ϑ (67) . Both V h V_h V h u ϑ u_\vartheta u ϑ max j v j \max_jv_j max j v j min { ∑ j v j , 1 } \min\{\sum_jv_j,1\} min { ∑ j v j , 1 } Theorem 3.4 therefore gives
V h ( t , x ) = u ϑ ( t , x ) for all ( t , x ) ∈ R × R N . V_h(t,x)=u_\vartheta(t,x)\qquad\text{for all }(t,x)\in\mathbb{R}\times\mathbb{R}^N. V h ( t , x ) = u ϑ ( t , x ) for all ( t , x ) ∈ R × R N . In particular, for each fixed r r r n n n V h n = u ϑ r , n V_{h_n}=u_{\vartheta_{r,n}} V h n = u ϑ r , n ϑ r , n \vartheta_{r,n} ϑ r , n X X X
Finally, choose r k ↗ 1 r_k\nearrow1 r k ↗ 1 μ ( B r k ‾ ) > 0 \mu(\overline{B_{r_k}})>0 μ ( B r k ) > 0
∥ V H r k − U ∥ C 1 , 2 ( [ − k , k ] × B k ‾ ) < 1 2 k . \|V_{H_{r_k}}-U\|_{C^{1,2}([-k,k]\times\overline{B_k})}<\frac{1}{2k}. ∥ V H r k − U ∥ C 1 , 2 ([ − k , k ] × B k ) < 2 k 1 . For each k k k n k n_k n k
∥ u ϑ r k , n k − V H r k ∥ C 1 , 2 ( [ − k , k ] × B k ‾ ) < 1 2 k . \|u_{\vartheta_{r_k,n_k}}-V_{H_{r_k}}\|_{C^{1,2}([-k,k]\times\overline{B_k})}<\frac{1}{2k}. ∥ u ϑ r k , n k − V H r k ∥ C 1 , 2 ([ − k , k ] × B k ) < 2 k 1 . Setting ϑ k = ϑ r k , n k \vartheta_k=\vartheta_{r_k,n_k} ϑ k = ϑ r k , n k
∥ u ϑ k − U ∥ C 1 , 2 ( [ − k , k ] × B k ‾ ) < 1 k , \|u_{\vartheta_k}-U\|_{C^{1,2}([-k,k]\times\overline{B_k})}<\frac{1}{k}, ∥ u ϑ k − U ∥ C 1 , 2 ([ − k , k ] × B k ) < k 1 , which proves the asserted convergence.
Small-amplitude limits under a strict gradient bound
We first identify the action of dilation on traveling fronts and the homogeneous solution and derive a scalar small-amplitude criterion. We then use it to prove that a strict logarithmic gradient bound gives a finite limit of D a U / a D_aU/a D a U / a
Throughout this subsection, U U U 0 < U < 1 0<U<1 0 < U < 1 Lemma 2.19 , and retain F 1 F_1 F 1 M M M (60) . The critical profile ϕ ∗ \phi_* ϕ ∗ (6) :
ϕ ∗ ( z ) = e λ ∗ z ( − z + O ( 1 ) ) as z → − ∞ . \phi_*(z)=e^{\lambda_*z}(-z+O(1))\quad\text{as }z\to-\infty. ϕ ∗ ( z ) = e λ ∗ z ( − z + O ( 1 )) as z → − ∞. Write q ( u ) = ϕ ∗ ′ ( ϕ ∗ − 1 ( u ) ) q(u)=\phi_*'(\phi_*^{-1}(u)) q ( u ) = ϕ ∗ ′ ( ϕ ∗ − 1 ( u )) 0 < u < 1 0<u<1 0 < u < 1 0 < q ( u ) < λ ∗ u 0<q(u)<\lambda_*u 0 < q ( u ) < λ ∗ u q ( u ) / u → λ ∗ q(u)/u\to\lambda_* q ( u ) / u → λ ∗ u ↘ 0 u\searrow0 u ↘ 0
The following lemma records the phase shifts produced by dilation and the critical-profile inequalities used in the measure construction.
Lemma 3.6. For every c ≥ c ∗ c\ge c_* c ≥ c ∗ ϕ c \phi_c ϕ c
0 < ϕ c ′ ( z ) ϕ c ( z ) < λ ( c ) for z ∈ R , ϕ c ′ ( z ) ϕ c ( z ) → λ ( c ) as z → − ∞ . 0<\frac{\phi_c'(z)}{\phi_c(z)}<\lambda(c)\quad\text{for }z\in\mathbb{R},\qquad\frac{\phi_c'(z)}{\phi_c(z)}\to\lambda(c)\quad\text{as }z\to-\infty. 0 < ϕ c ( z ) ϕ c ′ ( z ) < λ ( c ) for z ∈ R , ϕ c ( z ) ϕ c ′ ( z ) → λ ( c ) as z → − ∞. For every a > 0 a>0 a > 0 ν ∈ S N − 1 \nu\in\mathbb{S}^{N-1} ν ∈ S N − 1
D a [ ϕ c ( x ⋅ ν + c t ) ] = ϕ c ( x ⋅ ν + c t + λ ( c ) − 1 ln a ) . (75) D_a[\phi_c(x\cdot\nu+ct)]=\phi_c\left(x\cdot\nu+ct+\lambda(c)^{-1}\ln a\right). \tag*{(75)} D a [ ϕ c ( x ⋅ ν + c t )] = ϕ c ( x ⋅ ν + c t + λ ( c ) − 1 ln a ) . (75) The critical profile ϕ ∗ \phi_* ϕ ∗ ϕ ∗ ′ ′ − λ ∗ ϕ ∗ ′ ≤ 0 \phi_*''-\lambda_*\phi_*'\le0 ϕ ∗ ′′ − λ ∗ ϕ ∗ ′ ≤ 0 R \mathbb{R} R g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) g(q)=\phi_*(\lambda_*^{-1}\ln q) g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) q > 0 q>0 q > 0 g ( 0 ) = 0 g(0)=0 g ( 0 ) = 0 [ 0 , + ∞ ) [0,+\infty) [ 0 , + ∞ )
For the spatially homogeneous orbit ξ f \xi_f ξ f D a ξ f ( t ) = ξ f ( t + f ′ ( 0 ) − 1 ln a ) D_a\xi_f(t)=\xi_f(t+f'(0)^{-1}\ln a) D a ξ f ( t ) = ξ f ( t + f ′ ( 0 ) − 1 ln a )
Proof. The bounds on ϕ c ′ / ϕ c \phi_c'/\phi_c ϕ c ′ / ϕ c − ∞ -\infty − ∞ (4) –(7) . Integrating the logarithmic derivative gives
ln ϕ c ( z ) z → λ ( c ) , ϕ c ( z + b ) ϕ c ( z ) → e λ ( c ) b as z → − ∞ , (76) \frac{\ln\phi_c(z)}{z}\to\lambda(c),\qquad\frac{\phi_c(z+b)}{\phi_c(z)}\to e^{\lambda(c)b}\quad\text{as }z\to-\infty, \tag*{(76)} z ln ϕ c ( z ) → λ ( c ) , ϕ c ( z ) ϕ c ( z + b ) → e λ ( c ) b as z → − ∞ , (76) for every fixed b b b
For 0 < a < 1 0<a<1 0 < a < 1 λ ( c ) − 1 ln a < 0 \lambda(c)^{-1}\ln a<0 λ ( c ) − 1 ln a < 0 ϕ c ′ / ϕ c < λ ( c ) \phi_c'/\phi_c<\lambda(c) ϕ c ′ / ϕ c < λ ( c ) z + λ ( c ) − 1 ln a z+\lambda(c)^{-1}\ln a z + λ ( c ) − 1 ln a z z z
ln ϕ c ( z ) ϕ c ( z + λ ( c ) − 1 ln a ) = ∫ z + λ ( c ) − 1 ln a z ϕ c ′ ( s ) ϕ c ( s ) d s < − ln a . \ln\frac{\phi_c(z)}{\phi_c(z+\lambda(c)^{-1}\ln a)}
=\int_{z+\lambda(c)^{-1}\ln a}^{z}\frac{\phi_c'(s)}{\phi_c(s)}\,\mathrm{d}s<-\ln a. ln ϕ c ( z + λ ( c ) − 1 ln a ) ϕ c ( z ) = ∫ z + λ ( c ) − 1 l n a z ϕ c ( s ) ϕ c ′ ( s ) d s < − ln a . Thus a ϕ c ( z ) < ϕ c ( z + λ ( c ) − 1 ln a ) a\phi_c(z)<\phi_c(z+\lambda(c)^{-1}\ln a) a ϕ c ( z ) < ϕ c ( z + λ ( c ) − 1 ln a ) (41) yield
a ϕ c ( z ) ≤ D a ϕ c ( z ) ≤ ϕ c ( z + λ ( c ) − 1 ln a ) . a\phi_c(z)\le D_a\phi_c(z)\le\phi_c(z+\lambda(c)^{-1}\ln a). a ϕ c ( z ) ≤ D a ϕ c ( z ) ≤ ϕ c ( z + λ ( c ) − 1 ln a ) . Translation invariance of D a D_a D a ψ : = D a ϕ c \psi:=D_a\phi_c ψ := D a ϕ c
ψ ′ ′ − c ψ ′ + f ( ψ ) = 0 , 0 < ψ < 1. \psi''-c\psi'+f(\psi)=0,\qquad0<\psi<1. ψ ′′ − c ψ ′ + f ( ψ ) = 0 , 0 < ψ < 1. Since c > 0 c>0 c > 0 Theorem 2.1 to the entire solution ψ ( x ⋅ ν + c t ) \psi(x\cdot\nu+ct) ψ ( x ⋅ ν + c t )
ψ ′ > 0 , lim z → − ∞ ψ ( z ) = 0 , lim z → + ∞ ψ ( z ) = 1. \psi'>0,\qquad\lim_{z\to-\infty}\psi(z)=0,\qquad\lim_{z\to+\infty}\psi(z)=1. ψ ′ > 0 , z → − ∞ lim ψ ( z ) = 0 , z → + ∞ lim ψ ( z ) = 1. Uniqueness of traveling fronts therefore gives ψ ( z ) = ϕ c ( z + b ) \psi(z)=\phi_c(z+b) ψ ( z ) = ϕ c ( z + b ) b ∈ R b\in\mathbb{R} b ∈ R ϕ c ( z ) \phi_c(z) ϕ c ( z ) z → − ∞ z\to-\infty z → − ∞ (76) , we obtain
a ≤ lim z → − ∞ ψ ( z ) ϕ c ( z ) = e λ ( c ) b ≤ lim z → − ∞ ϕ c ( z + λ ( c ) − 1 ln a ) ϕ c ( z ) = a . a\leq\lim_{z\to-\infty}\frac{\psi(z)}{\phi_c(z)}=e^{\lambda(c)b}\leq\lim_{z\to-\infty}\frac{\phi_c(z+\lambda(c)^{-1}\ln a)}{\phi_c(z)}=a. a ≤ z → − ∞ lim ϕ c ( z ) ψ ( z ) = e λ ( c ) b ≤ z → − ∞ lim ϕ c ( z ) ϕ c ( z + λ ( c ) − 1 ln a ) = a . Thus b = λ ( c ) − 1 ln a b=\lambda(c)^{-1}\ln a b = λ ( c ) − 1 ln a (75) for 0 < a < 1 0<a<1 0 < a < 1 a = 1 a=1 a = 1 a > 1 a>1 a > 1 a − 1 < 1 a^{-1}<1 a − 1 < 1 D 1 / a [ ϕ c ( ⋅ + λ ( c ) − 1 ln a ) ] = ϕ c D_{1/a}[\phi_c(\cdot+\lambda(c)^{-1}\ln a)]=\phi_c D 1/ a [ ϕ c ( ⋅ + λ ( c ) − 1 ln a )] = ϕ c D a D_a D a D a ϕ c = ϕ c ( ⋅ + λ ( c ) − 1 ln a ) D_a\phi_c=\phi_c(\cdot+\lambda(c)^{-1}\ln a) D a ϕ c = ϕ c ( ⋅ + λ ( c ) − 1 ln a ) (75) for a > 1 a>1 a > 1
Finally put h = ϕ ∗ ′ ′ − λ ∗ ϕ ∗ ′ = λ ∗ ϕ ∗ ′ − f ( ϕ ∗ ) h=\phi_*''-\lambda_*\phi_*'=\lambda_*\phi_*'-f(\phi_*) h = ϕ ∗ ′′ − λ ∗ ϕ ∗ ′ = λ ∗ ϕ ∗ ′ − f ( ϕ ∗ )
h ′ − λ ∗ h = ( f ′ ( 0 ) − f ′ ( ϕ ∗ ) ) ϕ ∗ ′ ≥ 0. h'-\lambda_*h=(f'(0)-f'(\phi_*))\phi_*'\geq0. h ′ − λ ∗ h = ( f ′ ( 0 ) − f ′ ( ϕ ∗ )) ϕ ∗ ′ ≥ 0. The profile and its first two derivatives are bounded, so e − λ ∗ z h ( z ) → 0 e^{-\lambda_*z}h(z)\to0 e − λ ∗ z h ( z ) → 0 z → + ∞ z\to+\infty z → + ∞ ( e − λ ∗ z h ) ′ ≥ 0 (e^{-\lambda_*z}h)'\geq0 ( e − λ ∗ z h ) ′ ≥ 0 h ≤ 0 h\leq0 h ≤ 0 λ ∗ ϕ ∗ ′ ≤ f ( ϕ ∗ ) \lambda_*\phi_*'\leq f(\phi_*) λ ∗ ϕ ∗ ′ ≤ f ( ϕ ∗ ) q 2 g ′ ′ ( q ) = λ ∗ − 2 h ( λ ∗ − 1 ln q ) ≤ 0 q^2g''(q)=\lambda_*^{-2}h(\lambda_*^{-1}\ln q)\leq0 q 2 g ′′ ( q ) = λ ∗ − 2 h ( λ ∗ − 1 ln q ) ≤ 0 g g g ξ f \xi_f ξ f D a ξ f D_a\xi_f D a ξ f ξ f \xi_f ξ f a ξ f ( t ) ≤ D a ξ f ( t ) ≤ ξ f ( t + f ′ ( 0 ) − 1 ln a ) a\xi_f(t)\leq D_a\xi_f(t)\leq\xi_f(t+f'(0)^{-1}\ln a) a ξ f ( t ) ≤ D a ξ f ( t ) ≤ ξ f ( t + f ′ ( 0 ) − 1 ln a ) 0 < a < 1 0<a<1 0 < a < 1 f ( u ) / u ≤ f ′ ( 0 ) f(u)/u\leq f'(0) f ( u ) / u ≤ f ′ ( 0 ) e f ′ ( 0 ) t e^{f'(0)t} e f ′ ( 0 ) t f ′ ( 0 ) − 1 ln a f'(0)^{-1}\ln a f ′ ( 0 ) − 1 ln a a > 0 a>0 a > 0
We next turn the amplitude differential inequality into a scalar criterion for the limits at scales a a a a ln ( 1 / a ) a\ln(1/a) a ln ( 1/ a ) G ( a ) = D a U ( t , x ) G(a)=D_aU(t,x) G ( a ) = D a U ( t , x )
Lemma 3.7. Let G ∈ C 1 ( ( 0 , ∞ ) ; ( 0 , 1 ) ) G\in C^1((0,\infty);(0,1)) G ∈ C 1 (( 0 , ∞ ) ; ( 0 , 1 )) lim a ↓ 0 G ( a ) = 0 \lim_{a\downarrow0}G(a)=0 lim a ↓ 0 G ( a ) = 0 G − a G ′ ≥ 0 G-aG'\geq0 G − a G ′ ≥ 0 0 < γ ≤ 1 0<\gamma\leq1 0 < γ ≤ 1 M > 0 M>0 M > 0
γ a 2 G ′ ′ ≤ a G ′ − G + M G 2 in distributions on ( 0 , ∞ ) . (77) \gamma a^2G''\leq aG'-G+MG^2\qquad\text{in distributions on }(0,\infty). \tag*{(77)} γ a 2 G ′′ ≤ a G ′ − G + M G 2 in distributions on ( 0 , ∞ ) . (77) If γ = 1 \gamma=1 γ = 1 A A A
lim a ↓ 0 G ( a ) a ln ( 1 / a ) = A . \lim_{a\downarrow0}\frac{G(a)}{a\ln(1/a)}=A. a ↓ 0 lim a ln ( 1/ a ) G ( a ) = A . If 0 < γ < 1 0<\gamma<1 0 < γ < 1 A A A
lim a ↓ 0 G ( a ) a = A . \lim_{a\downarrow0}\frac{G(a)}{a}=A. a ↓ 0 lim a G ( a ) = A . Proof. Set a = e − τ a=e^{-\tau} a = e − τ G ~ ( τ ) = e τ G ( e − τ ) \widetilde G(\tau)=e^\tau G(e^{-\tau}) G ( τ ) = e τ G ( e − τ )
G ~ ′ = G / a − G ′ ≥ 0 , G ~ ′ ′ = G ~ ′ + a G ′ ′ , \widetilde G'=G/a-G'\geq0,\qquad\widetilde G''=\widetilde G'+aG'', G ′ = G / a − G ′ ≥ 0 , G ′′ = G ′ + a G ′′ , where the second identity holds in distributions. Since G G G G ′ G' G ′ G ~ ′ \widetilde G' G ′
G ~ ′ ′ + ( γ − 1 − 1 ) G ~ ′ ≤ M γ e − τ G ~ 2 . (78) \widetilde G''+(\gamma^{-1}-1)\widetilde G'\leq\frac{M}{\gamma}e^{-\tau}\widetilde G^2. \tag*{(78)} G ′′ + ( γ − 1 − 1 ) G ′ ≤ γ M e − τ G 2 . (78) Since G ~ > 0 \widetilde G>0 G > 0 γ ≤ 1 \gamma\leq1 γ ≤ 1
( G ~ ′ G ~ ) ′ ≤ M γ G ( e − τ ) − ( G ~ ′ G ~ ) 2 . \left(\frac{\widetilde G'}{\widetilde G}\right)'\leq\frac{M}{\gamma}G(e^{-\tau})-\left(\frac{\widetilde G'}{\widetilde G}\right)^2. ( G G ′ ) ′ ≤ γ M G ( e − τ ) − ( G G ′ ) 2 . Since G ~ ( e − τ ) → 0 \widetilde{G}(e^{-\tau}) \to0 G ( e − τ ) → 0 τ → + ∞ \tau\to+\infty τ → + ∞ ε > 0 \varepsilon> 0 ε > 0 τ ε \tau_\varepsilon τ ε ( M / γ ) G ~ ( e − τ ) ≤ ε 2 (M/\gamma)\widetilde{G}(e^{-\tau}) \le\varepsilon^2 ( M / γ ) G ( e − τ ) ≤ ε 2 τ ≥ τ ε \tau\ge\tau_\varepsilon τ ≥ τ ε ε + 1 / ( τ − τ ε ) \varepsilon+ 1/(\tau-\tau_\varepsilon) ε + 1/ ( τ − τ ε )
0 ≤ G ~ ′ ( τ ) G ~ ( τ ) ≤ ε + 1 τ − τ ε for τ > τ ε . 0 \le\frac{\widetilde{G}'(\tau)}{\widetilde{G}(\tau)} \le\varepsilon+\frac{1}{\tau-\tau_\varepsilon} \qquad\text{for }\tau>\tau_\varepsilon. 0 ≤ G ( τ ) G ′ ( τ ) ≤ ε + τ − τ ε 1 for τ > τ ε . The comparison remains valid in distributions: the singular part of the derivative is nonpositive, and the positive integrating factor for the difference preserves its sign. Consequently G ~ ′ / G ~ → 0 \widetilde{G}'/\widetilde{G} \to0 G ′ / G → 0 τ 0 \tau_0 τ 0 G ~ ′ / G ~ ≤ 1 / 4 \widetilde{G}'/\widetilde{G} \le1/4 G ′ / G ≤ 1/4 τ ≥ τ 0 \tau\ge\tau_0 τ ≥ τ 0 (78) give, for some C > 0 C>0 C > 0
G ~ ( τ ) ≤ G ~ ( τ 0 ) e ( τ − τ 0 ) / 4 , G ~ ′ ′ + ( γ − 1 − 1 ) G ~ ′ ≤ C e − τ / 2 . \widetilde{G}(\tau) \le\widetilde{G}(\tau_0)e^{(\tau-\tau_0)/4}, \qquad\widetilde{G}''+(\gamma^{-1}-1)\widetilde{G}' \le Ce^{-\tau/2}. G ( τ ) ≤ G ( τ 0 ) e ( τ − τ 0 ) /4 , G ′′ + ( γ − 1 − 1 ) G ′ ≤ C e − τ /2 . Hence
d d τ ( G ~ ′ + 2 C e − τ / 2 ) ≤ − ( γ − 1 − 1 ) G ~ ′ ≤ 0. \frac{d}{d\tau}\left(\widetilde{G}'+2Ce^{-\tau/2}\right) \le-(\gamma^{-1}-1)\widetilde{G}' \le0. d τ d ( G ′ + 2 C e − τ /2 ) ≤ − ( γ − 1 − 1 ) G ′ ≤ 0. The nonnegative function G ~ ′ + 2 C e − τ / 2 \widetilde{G}'+2Ce^{-\tau/2} G ′ + 2 C e − τ /2 G ~ ′ \widetilde{G}' G ′ γ = 1 \gamma=1 γ = 1 A = lim τ → ∞ G ~ ′ ( τ ) ≥ 0 A=\lim_{\tau\to\infty}\widetilde{G}'(\tau)\ge0 A = lim τ → ∞ G ′ ( τ ) ≥ 0
G ( a ) a ln ( 1 / a ) = G ~ ( τ ) τ = G ~ ( τ 0 ) τ + 1 τ ∫ τ 0 τ G ~ ′ ( s ) d s → A . \frac{G(a)}{a\ln(1/a)}=\frac{\widetilde{G}(\tau)}{\tau}=\frac{\widetilde{G}(\tau_0)}{\tau}+\frac{1}{\tau}\int_{\tau_0}^{\tau}\widetilde{G}'(s)\,ds\to A. a ln ( 1/ a ) G ( a ) = τ G ( τ ) = τ G ( τ 0 ) + τ 1 ∫ τ 0 τ G ′ ( s ) d s → A . If 0 < γ < 1 0<\gamma<1 0 < γ < 1
( γ − 1 − 1 ) ∫ τ 0 τ G ~ ′ ( s ) d s ≤ G ~ ( τ 0 ) + 2 C e − τ 0 / 2 . (\gamma^{-1}-1)\int_{\tau_0}^{\tau}\widetilde{G}'(s)\,ds\le\widetilde{G}(\tau_0)+2Ce^{-\tau_0/2}. ( γ − 1 − 1 ) ∫ τ 0 τ G ′ ( s ) d s ≤ G ( τ 0 ) + 2 C e − τ 0 /2 . Thus G ~ \widetilde{G} G A ∈ ( 0 , ∞ ) A\in(0,\infty) A ∈ ( 0 , ∞ )
G ( a ) a = G ~ ( τ ) → A as τ → ∞ or, equivalently, a ↘ 0. \frac{G(a)}{a}=\widetilde{G}(\tau)\to A \qquad\text{as }\tau\to\infty\text{ or, equivalently, }a\searrow0. a G ( a ) = G ( τ ) → A as τ → ∞ or, equivalently, a ↘ 0. This completes the proof.
We now give a sufficient condition for (68) : a strict logarithmic gradient bound ensures that D a U / a D_aU/a D a U / a
Theorem 3.8. If ∣ ∇ ln U ∣ ≤ λ ∗ r |\nabla\ln U|\le\lambda_*r ∣∇ ln U ∣ ≤ λ ∗ r 0 ≤ r < 1 0\le r<1 0 ≤ r < 1
H ( t , x ) : = lim T → ∞ [ L t + T [ U ( − T ) ] ] ( x ) = lim a ↘ 0 D a U ( t , x ) a < ∞ . (79) H(t,x):=\lim_{T\to\infty}[L_{t+T}[U(-T)]](x)=\lim_{a\searrow0}\frac{D_aU(t,x)}{a}<\infty. \tag*{(79)} H ( t , x ) := T → ∞ lim [ L t + T [ U ( − T )]] ( x ) = a ↘ 0 lim a D a U ( t , x ) < ∞. (79) Both limits hold in C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) μ \mu μ B r ‾ \overline{B_r} B r
H ( t , x ) = ∫ B r ‾ e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t μ ( d ξ ) . H(t,x)=\int_{\overline{B_r}}e^{\lambda_*x\cdot\xi+f'(0)(1+|\xi|^2)t}\,\mu(d\xi). H ( t , x ) = ∫ B r e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t μ ( d ξ ) . In particular, Theorem 3.5 reconstructs U U U
Proof. Set γ = ( 1 + r 2 ) / 2 \gamma=(1+r^2)/2 γ = ( 1 + r 2 ) /2 r 2 / 2 ≤ γ < 1 r^2/2\le\gamma<1 r 2 /2 ≤ γ < 1 γ > 0 \gamma>0 γ > 0 r = 0 r=0 r = 0 G ( a ) = D a U ( t , x ) G(a)=D_aU(t,x) G ( a ) = D a U ( t , x ) (42) and f ( G ) / f ′ ( 0 ) ≥ G − ( M / f ′ ( 0 ) ) G 2 f(G)/f'(0)\ge G-(M/f'(0))G^2 f ( G ) / f ′ ( 0 ) ≥ G − ( M / f ′ ( 0 )) G 2 (77) with this γ \gamma γ M / f ′ ( 0 ) M/f'(0) M / f ′ ( 0 )
H 0 ( t , x ) = lim a ↘ 0 D a U ( t , x ) a . H_0(t,x)=\lim_{a\searrow0}\frac{D_aU(t,x)}{a}. H 0 ( t , x ) = a ↘ 0 lim a D a U ( t , x ) . Set Z a = D a U / a Z_a = D_aU/a Z a = D a U / a Theorem 2.1 to D a U D_aU D a U
e − 2 f ′ ( 0 ) ∣ t ∣ − λ ∗ ∣ x ∣ Z a ( 0 , 0 ) ≤ Z a ( t , x ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ + λ ∗ ∣ x ∣ Z a ( 0 , 0 ) . e^{-2f'(0)|t|-\lambda_*|x|}Z_a(0,0) \le Z_a(t,x) \le e^{2f'(0)|t|+\lambda_*|x|}Z_a(0,0). e − 2 f ′ ( 0 ) ∣ t ∣ − λ ∗ ∣ x ∣ Z a ( 0 , 0 ) ≤ Z a ( t , x ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ + λ ∗ ∣ x ∣ Z a ( 0 , 0 ) . Since Z a ( 0 , 0 ) → H 0 ( 0 , 0 ) < ∞ Z_a(0,0) \to H_0(0,0) < \infty Z a ( 0 , 0 ) → H 0 ( 0 , 0 ) < ∞ a → 0 a \to0 a → 0 Z a Z_a Z a a a a
( Z a ) t − Δ Z a = f ′ ( 0 ) Z a − F 1 ( a Z a ) a , 0 ≤ F 1 ( a Z a ) a ≤ M a Z a 2 , (Z_a)_t-\Delta Z_a=f'(0)Z_a-\frac{F_1(aZ_a)}{a}, \qquad0\le\frac{F_1(aZ_a)}{a}\le MaZ_a^2, ( Z a ) t − Δ Z a = f ′ ( 0 ) Z a − a F 1 ( a Z a ) , 0 ≤ a F 1 ( a Z a ) ≤ M a Z a 2 , interior parabolic estimates give convergence in C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2
( H 0 ) t = Δ H 0 + f ′ ( 0 ) H 0 , ∣ ∇ H 0 ∣ ≤ λ ∗ r H 0 . (H_0)_t=\Delta H_0+f'(0)H_0,\qquad|\nabla H_0|\le\lambda_*rH_0. ( H 0 ) t = Δ H 0 + f ′ ( 0 ) H 0 , ∣∇ H 0 ∣ ≤ λ ∗ r H 0 . The gradient bound follows from Lemma 2.3 . The representation (17) , together with ∣ ∇ H 0 ∣ ≤ λ ∗ r H 0 |\nabla H_0|\le\lambda_*rH_0 ∣∇ H 0 ∣ ≤ λ ∗ r H 0 B ‾ r \overline{B}_r B r Theorem 3.5 with radius 1 1 1 r r r
For t > − T t>-T t > − T H T ( t ) = L t + T [ U ( − T ) ] H_T(t)=L_{t+T}[U(-T)] H T ( t ) = L t + T [ U ( − T )] F 1 ≥ 0 F_1\ge0 F 1 ≥ 0
∂ T H T ( t ) = L t + T [ F 1 ( U ( − T ) ) ] ≥ 0. \partial_T H_T(t)=L_{t+T}[F_1(U(-T))]\ge0. ∂ T H T ( t ) = L t + T [ F 1 ( U ( − T ))] ≥ 0. Also D a U ≥ a U D_aU\ge aU D a U ≥ a U 0 < a ≤ 1 0<a\le1 0 < a ≤ 1 U ≤ H 0 U\le H_0 U ≤ H 0
H T ( t ) ≤ L t + T [ H 0 ( − T ) ] = H 0 ( t ) . H_T(t)\le L_{t+T}[H_0(-T)]=H_0(t). H T ( t ) ≤ L t + T [ H 0 ( − T )] = H 0 ( t ) . The last equality follows either from the spectral representation or from Gaussian integration. Consequently H T ↗ H ≤ H 0 H_T \nearrow H\le H_0 H T ↗ H ≤ H 0
S t + T [ a U ( − T ) ] ≤ a L t + T [ U ( − T ) ] . S_{t+T}[aU(-T)]\le aL_{t+T}[U(-T)]. S t + T [ a U ( − T )] ≤ a L t + T [ U ( − T )] . First letting T → ∞ T\to\infty T → ∞ a ↘ 0 a\searrow0 a ↘ 0 D a U ≤ a H D_aU\le aH D a U ≤ a H H 0 ≤ H H_0\le H H 0 ≤ H H = H 0 H=H_0 H = H 0 H T → H H_T\to H H T → H C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2 Theorem 3.5 .
The strict logarithmic gradient bound above is sufficient but is not assumed for a general entire solution. In § 8 , we decompose such a solution into a critical part and an increasing limit of solutions satisfying strict logarithmic gradient bounds. The latter are covered by the reconstruction established here. For the critical part, § 9.3 uses the different normalization D a U / [ a ln ( 1 / a ) ] D_aU/[a\ln(1/a)] D a U / [ a ln ( 1/ a )]
Translation parameters and critical–subcritical decomposition
§ 6 constructs the ordered vector space E E E E U E_U E U Proposition 2.24 represents space–time translations on E U E_U E U e s λ ( k ) + λ ∗ h ⋅ ξ ( k ) e^{s\lambda(k)+\lambda_*h\cdot\xi(k)} e s λ ( k ) + λ ∗ h ⋅ ξ ( k ) λ ( k ) = f ′ ( 0 ) ( 1 + ∣ ξ ( k ) ∣ 2 ) \lambda(k)=f'(0)(1+|\xi(k)|^2) λ ( k ) = f ′ ( 0 ) ( 1 + ∣ ξ ( k ) ∣ 2 )
Identification of the translation parameters
We show that the translation parameters satisfy λ = f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) \lambda= f'(0)(1 + |\xi|^2) λ = f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 )
Fix a nonzero U ∈ C U \in\mathcal{C} U ∈ C K U K_U K U
J : E U → C ( K U ) , J ( U ) = 1 , \mathcal{J}: E_U \to C(K_U), \qquad\mathcal{J}(U)=1, J : E U → C ( K U ) , J ( U ) = 1 , and continuous functions
λ : K U → [ 0 , 2 f ′ ( 0 ) ] , ξ : K U → B ‾ 1 \lambda: K_U \to[0,2f'(0)], \qquad\xi: K_U \to\overline{B}_1 λ : K U → [ 0 , 2 f ′ ( 0 )] , ξ : K U → B 1 such that
J ( T s , h Z ) ( k ) = e s λ ( k ) + λ ∗ h ⋅ ξ ( k ) J ( Z ) ( k ) for Z ∈ E U . (80) \mathcal{J}(T_{s,h}Z)(k)=e^{s\lambda(k)+\lambda_*h\cdot\xi(k)}\mathcal{J}(Z)(k) \qquad\text{for } Z\in E_U. \tag*{(80)} J ( T s , h Z ) ( k ) = e s λ ( k ) + λ ∗ h ⋅ ξ ( k ) J ( Z ) ( k ) for Z ∈ E U . (80) Only the bounds 0 ≤ λ ≤ 2 f ′ ( 0 ) 0 \leq\lambda\leq2f'(0) 0 ≤ λ ≤ 2 f ′ ( 0 ) ∣ ξ ∣ ≤ 1 |\xi| \leq1 ∣ ξ ∣ ≤ 1 λ ≥ f ′ ( 0 ) \lambda\geq f'(0) λ ≥ f ′ ( 0 )
We now determine the relation imposed by the equation on the temporal and spatial translation parameters.
Theorem 4.1. For every k ∈ K U k \in K_U k ∈ K U
λ ( k ) = f ′ ( 0 ) ( 1 + ∣ ξ ( k ) ∣ 2 ) . (81) \lambda(k)=f'(0)(1+|\xi(k)|^2). \tag*{(81)} λ ( k ) = f ′ ( 0 ) ( 1 + ∣ ξ ( k ) ∣ 2 ) . (81) Proof. Fix k 0 ∈ K U k_0 \in K_U k 0 ∈ K U λ 0 = λ ( k 0 ) \lambda_0=\lambda(k_0) λ 0 = λ ( k 0 ) ξ 0 = ξ ( k 0 ) \xi_0=\xi(k_0) ξ 0 = ξ ( k 0 ) n ≥ 1 n \geq1 n ≥ 1 ψ n ∈ C ( K U ) \psi_n \in C(K_U) ψ n ∈ C ( K U )
0 ≤ ψ n ≤ 1 , ψ n ( k 0 ) = 1 , supp ψ n ⊂ { k : ∣ λ ( k ) − λ 0 ∣ + ∣ ξ ( k ) − ξ 0 ∣ < 1 / n } . 0\leq\psi_n\leq1,\qquad\psi_n(k_0)=1,\qquad\operatorname{supp}\psi_n\subset\{k:|\lambda(k)-\lambda_0|+|\xi(k)-\xi_0|<1/n\}. 0 ≤ ψ n ≤ 1 , ψ n ( k 0 ) = 1 , supp ψ n ⊂ { k : ∣ λ ( k ) − λ 0 ∣ + ∣ ξ ( k ) − ξ 0 ∣ < 1/ n } . Set Z n = J − 1 ψ n Z_n=\mathcal{J}^{-1}\psi_n Z n = J − 1 ψ n 0 ≤ Z n ≤ U < 1 0 \leq Z_n \leq U < 1 0 ≤ Z n ≤ U < 1 ψ n ( k 0 ) = 1 \psi_n(k_0)=1 ψ n ( k 0 ) = 1 0 < Z n < 1 0<Z_n<1 0 < Z n < 1 θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) b n > 0 b_n>0 b n > 0
Y n : = D b n Z n , Y n ( 0 , 0 ) = θ . Y_n:=D_{b_n}Z_n,\qquad Y_n(0,0)=\theta. Y n := D b n Z n , Y n ( 0 , 0 ) = θ . Moreover, J ( Y n ) = b n ψ n \mathcal{J}(Y_n)=b_n\psi_n J ( Y n ) = b n ψ n (51) , for 0 < a < b 0<a<b 0 < a < b
0 ≤ D b Y − D a Y = D b / a ( D a Y ) − D a Y ≤ ( b a − 1 ) D a Y ≤ b − a a . (82) 0\leq D_bY-D_aY=D_{b/a}(D_aY)-D_aY\leq\left(\frac{b}{a}-1\right)D_aY\leq\frac{b-a}{a}. \tag*{(82)} 0 ≤ D b Y − D a Y = D b / a ( D a Y ) − D a Y ≤ ( a b − 1 ) D a Y ≤ a b − a . (82) Interior parabolic estimates, (82) , and diagonal extraction give, along a subsequence,
D a Y n ( t , x ) ⟶ A ( a ; t , x ) D_aY_n(t,x)\longrightarrow\mathcal{A}(a;t,x) D a Y n ( t , x ) ⟶ A ( a ; t , x ) locally uniformly on ( 0 , ∞ ) × R × R N (0,\infty)\times\mathbb{R}\times\mathbb{R}^N ( 0 , ∞ ) × R × R N
For fixed a > 0 a>0 a > 0 C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2 ( t , x ) (t,x) ( t , x ) A ( a ; ⋅ , ⋅ ) \mathcal{A}(a;\cdot,\cdot) A ( a ; ⋅ , ⋅ ) (1) . In particular,
Y : = A ( 1 ; ⋅ , ⋅ ) , Y ( 0 , 0 ) = θ , Y:=\mathcal{A}(1;\cdot,\cdot),\qquad Y(0,0)=\theta, Y := A ( 1 ; ⋅ , ⋅ ) , Y ( 0 , 0 ) = θ , is an entire solution of (1) satisfying 0 < Y < 1 0<Y<1 0 < Y < 1
Put d n = ( ∣ t ∣ + λ ∗ ∣ x ∣ ) / n d_n=(|t|+\lambda_*|x|)/n d n = ( ∣ t ∣ + λ ∗ ∣ x ∣ ) / n (80) on the support of ψ n \psi_n ψ n ( 0 , 0 ) (0,0) ( 0 , 0 )
D a e λ 0 t + λ ∗ ξ 0 ⋅ x − d n Y n ( 0 , 0 ) ≤ D a Y n ( t , x ) ≤ D a e λ 0 t + λ ∗ ξ 0 ⋅ x + d n Y n ( 0 , 0 ) . D_a e^{\lambda_0t+\lambda_*\xi_0\cdot x-d_n}Y_n(0,0)
\leq D_aY_n(t,x)
\leq D_a e^{\lambda_0t+\lambda_*\xi_0\cdot x+d_n}Y_n(0,0). D a e λ 0 t + λ ∗ ξ 0 ⋅ x − d n Y n ( 0 , 0 ) ≤ D a Y n ( t , x ) ≤ D a e λ 0 t + λ ∗ ξ 0 ⋅ x + d n Y n ( 0 , 0 ) . By (82) and (4.4),
A ( a ; t , x ) = F ( a e λ 0 t + λ ∗ ξ 0 ⋅ x ) , F ( a ) : = A ( a ; 0 , 0 ) , F ( 1 ) = θ . (83) \mathcal{A}(a;t,x)=F(ae^{\lambda_0t+\lambda_*\xi_0\cdot x}),\qquad F(a):=\mathcal{A}(a;0,0),\qquad F(1)=\theta. \tag*{(83)} A ( a ; t , x ) = F ( a e λ 0 t + λ ∗ ξ 0 ⋅ x ) , F ( a ) := A ( a ; 0 , 0 ) , F ( 1 ) = θ . (83) By Theorem 2.1, Y t > 0 Y_t>0 Y t > 0 λ 0 = 0 \lambda_0=0 λ 0 = 0 Y t = 0 Y_t=0 Y t = 0 λ 0 > 0 \lambda_0>0 λ 0 > 0
F ( a ) = Y ( ln a λ 0 , 0 ) , F ′ ( a ) = Y t ( ln a λ 0 , 0 ) λ 0 a > 0 , 0 < F ( a ) < 1. F(a)=Y\left(\frac{\ln a}{\lambda_0},0\right),\qquad F'(a)=\frac{Y_t\left(\frac{\ln a}{\lambda_0},0\right)}{\lambda_0a}>0,\qquad0<F(a)<1. F ( a ) = Y ( λ 0 ln a , 0 ) , F ′ ( a ) = λ 0 a Y t ( λ 0 l n a , 0 ) > 0 , 0 < F ( a ) < 1. Passing to the limit in (43) and (44) gives
F ( a ) ≤ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( θ ) ) for 0 < a ≤ 1 , F ( a ) ≥ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( θ ) ) for a ≥ 1. F(a)\leq\phi_* \left(\lambda_*^{-1}\ln a+\phi_*^{-1}(\theta)\right)\quad\text{for }0<a\leq1,\qquad F(a)\geq\phi_* \left(\lambda_*^{-1}\ln a+\phi_*^{-1}(\theta)\right)\quad\text{for }a\geq1. F ( a ) ≤ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( θ ) ) for 0 < a ≤ 1 , F ( a ) ≥ ϕ ∗ ( λ ∗ − 1 ln a + ϕ ∗ − 1 ( θ ) ) for a ≥ 1. In particular,
lim a ↘ 0 F ( a ) = 0 , lim a → ∞ F ( a ) = 1 , F ( a ) ≤ C θ a ln ( 1 / a ) for 0 < a ≤ a θ , (84) \lim_{a\searrow0}F(a)=0,\qquad\lim_{a\to\infty}F(a)=1,\qquad F(a)\leq C_\theta a\ln(1/a)\quad\text{for }0<a\leq a_\theta, \tag*{(84)} a ↘ 0 lim F ( a ) = 0 , a → ∞ lim F ( a ) = 1 , F ( a ) ≤ C θ a ln ( 1/ a ) for 0 < a ≤ a θ , (84) for some C θ > 0 C_\theta>0 C θ > 0 a θ ∈ ( 0 , 1 ) a_\theta\in(0,1) a θ ∈ ( 0 , 1 ) (5) . We also have
D a Y ( 0 , 0 ) ≤ F ( a ) for 0 < a < 1. (85) D_aY(0,0)\leq F(a)\qquad\text{for }0<a<1. \tag*{(85)} D a Y ( 0 , 0 ) ≤ F ( a ) for 0 < a < 1. (85) Indeed, (39) gives, for fixed T > 0 T>0 T > 0
∣ S T [ a Y n ( − T ) ] − S T [ a Y ( − T ) ] ∣ ( 0 ) ≤ a e f ′ ( 0 ) T e T Δ ∣ Y n ( − T ) − Y ( − T ) ∣ ( 0 ) → 0. \left|S_T[aY_n(-T)]-S_T[aY(-T)]\right|(0)\leq ae^{f'(0)T}e^{T\Delta}|Y_n(-T)-Y(-T)|(0)\to0. ∣ S T [ a Y n ( − T )] − S T [ aY ( − T )] ∣ ( 0 ) ≤ a e f ′ ( 0 ) T e T Δ ∣ Y n ( − T ) − Y ( − T ) ∣ ( 0 ) → 0. For 0 < a < 1 0<a<1 0 < a < 1 T T T
D a Y ( 0 , 0 ) = sup T > 0 lim n → ∞ ( S T [ a Y n ( − T ) ] ) ( 0 ) ≤ lim inf n → ∞ D a Y n ( 0 , 0 ) = F ( a ) . D_aY(0,0)=\sup_{T>0}\lim_{n\to\infty}\left(S_T[aY_n(-T)]\right)(0)\leq\liminf_{n\to\infty}D_aY_n(0,0)=F(a). D a Y ( 0 , 0 ) = T > 0 sup n → ∞ lim ( S T [ a Y n ( − T )] ) ( 0 ) ≤ n → ∞ lim inf D a Y n ( 0 , 0 ) = F ( a ) . Suppose first that ξ 0 = 0 \xi_0=0 ξ 0 = 0 (83) into the equation gives
λ 0 a F ′ ( a ) = f ( F ( a ) ) . \lambda_0aF'(a)=f(F(a)). λ 0 a F ′ ( a ) = f ( F ( a )) . Since f ( u ) = f ′ ( 0 ) u + O ( u 2 ) f(u)=f'(0)u+O(u^2) f ( u ) = f ′ ( 0 ) u + O ( u 2 )
F ( a ) = c 0 a f ′ ( 0 ) / λ 0 ( 1 + o ( 1 ) ) as a ↘ 0 , c 0 > 0. F(a)=c_0a^{f'(0)/\lambda_0}(1+o(1))\quad\text{as }a\searrow0,\qquad c_0>0. F ( a ) = c 0 a f ′ ( 0 ) / λ 0 ( 1 + o ( 1 )) as a ↘ 0 , c 0 > 0. Here 1 / f ( u ) − 1 / ( f ′ ( 0 ) u ) = O ( 1 ) 1/f(u)-1/(f'(0)u)=O(1) 1/ f ( u ) − 1/ ( f ′ ( 0 ) u ) = O ( 1 ) u → 0 u\to0 u → 0 (84) ,
c 0 a f ′ ( 0 ) / λ 0 ( 1 + o ( 1 ) ) ≤ C θ a ln ( 1 / a ) ⟹ λ 0 ≤ f ′ ( 0 ) . c_0a^{f'(0)/\lambda_0}(1+o(1))\leq C_\theta a\ln(1/a)\quad\Longrightarrow\quad\lambda_0\leq f'(0). c 0 a f ′ ( 0 ) / λ 0 ( 1 + o ( 1 )) ≤ C θ a ln ( 1/ a ) ⟹ λ 0 ≤ f ′ ( 0 ) . On the other hand, the homogeneous-orbit amplitude action gives
D a Y ( 0 , 0 ) = Y ( ln a f ′ ( 0 ) , 0 ) = F ( a λ 0 / f ′ ( 0 ) ) . D_aY(0,0)=Y\left(\frac{\ln a}{f'(0)},0\right)=F\left(a^{\lambda_0/f'(0)}\right). D a Y ( 0 , 0 ) = Y ( f ′ ( 0 ) ln a , 0 ) = F ( a λ 0 / f ′ ( 0 ) ) . Using (85) and F ′ > 0 F'>0 F ′ > 0
F ( a λ 0 / f ′ ( 0 ) ) ≤ F ( a ) ⟹ a λ 0 / f ′ ( 0 ) ≤ a ⟹ λ 0 ≥ f ′ ( 0 ) for 0 < a < 1. F\left(a^{\lambda_0/f'(0)}\right)\leq F(a)\quad\Longrightarrow\quad a^{\lambda_0/f'(0)}\leq a\quad\Longrightarrow\quad\lambda_0\geq f'(0)\qquad\text{for }0<a<1. F ( a λ 0 / f ′ ( 0 ) ) ≤ F ( a ) ⟹ a λ 0 / f ′ ( 0 ) ≤ a ⟹ λ 0 ≥ f ′ ( 0 ) for 0 < a < 1. Hence λ 0 = f ′ ( 0 ) \lambda_0=f'(0) λ 0 = f ′ ( 0 )
Suppose now that ξ 0 ≠ 0 \xi_0\neq0 ξ 0 = 0
ν 0 = ξ 0 ∣ ξ 0 ∣ , c = λ 0 λ ∗ ∣ ξ 0 ∣ , Ψ ( z ) = F ( e λ ∗ ∣ ξ 0 ∣ z ) . \nu_0=\frac{\xi_0}{|\xi_0|},\qquad c=\frac{\lambda_0}{\lambda_*|\xi_0|},\qquad\Psi(z)=F\left(e^{\lambda_*|\xi_0|z}\right). ν 0 = ∣ ξ 0 ∣ ξ 0 , c = λ ∗ ∣ ξ 0 ∣ λ 0 , Ψ ( z ) = F ( e λ ∗ ∣ ξ 0 ∣ z ) . Then
Y ( t , x ) = Ψ ( x ⋅ ν 0 + c t ) , Ψ ′ > 0 , Ψ ( − ∞ ) = 0 , Ψ ( + ∞ ) = 1. Y(t,x)=\Psi(x\cdot\nu_0+ct),\qquad\Psi'>0,\qquad\Psi(-\infty)=0,\qquad\Psi(+\infty)=1. Y ( t , x ) = Ψ ( x ⋅ ν 0 + c t ) , Ψ ′ > 0 , Ψ ( − ∞ ) = 0 , Ψ ( + ∞ ) = 1. Since Ψ ( z ) = Y ( 0 , z ν 0 ) \Psi(z)=Y(0,z\nu_0) Ψ ( z ) = Y ( 0 , z ν 0 ) Lemma 3.6 gives c ≥ c ∗ c\ge c_* c ≥ c ∗
r = c − c 2 − 4 f ′ ( 0 ) 2 ∈ ( 0 , λ ∗ ] , r=\frac{c-\sqrt{c^2-4f'(0)}}{2}\in(0,\lambda_*], r = 2 c − c 2 − 4 f ′ ( 0 ) ∈ ( 0 , λ ∗ ] , Ψ ′ ( z ) / Ψ ( z ) → r \Psi'(z)/\Psi(z)\to r Ψ ′ ( z ) /Ψ ( z ) → r z → − ∞ z\to-\infty z → − ∞
lim a → 0 ln F ( a ) ln a = r λ ∗ ∣ ξ 0 ∣ . \lim_{a\to0}\frac{\ln F(a)}{\ln a}=\frac{r}{\lambda_*|\xi_0|}. a → 0 lim ln a ln F ( a ) = λ ∗ ∣ ξ 0 ∣ r . The upper bound in (84) implies r ≥ λ ∗ ∣ ξ 0 ∣ r\ge\lambda_*|\xi_0| r ≥ λ ∗ ∣ ξ 0 ∣ (85) give
Ψ ( r − 1 ln a ) = D a Y ( 0 , 0 ) ≤ F ( a ) = Ψ ( ( λ ∗ ∣ ξ 0 ∣ ) − 1 ln a ) for 0 < a < 1. \Psi(r^{-1}\ln a)=D_aY(0,0)\le F(a)=\Psi((\lambda_*|\xi_0|)^{-1}\ln a)\qquad\text{for }0<a<1. Ψ ( r − 1 ln a ) = D a Y ( 0 , 0 ) ≤ F ( a ) = Ψ (( λ ∗ ∣ ξ 0 ∣ ) − 1 ln a ) for 0 < a < 1. Since Ψ \Psi Ψ ln a < 0 \ln a<0 ln a < 0 r ≤ λ ∗ ∣ ξ 0 ∣ r\le\lambda_*|\xi_0| r ≤ λ ∗ ∣ ξ 0 ∣ r = λ ∗ ∣ ξ 0 ∣ r=\lambda_*|\xi_0| r = λ ∗ ∣ ξ 0 ∣ r 2 − c r + f ′ ( 0 ) = 0 r^2-cr+f'(0)=0 r 2 − cr + f ′ ( 0 ) = 0
λ 0 = c λ ∗ ∣ ξ 0 ∣ = f ′ ( 0 ) ( 1 + ∣ ξ 0 ∣ 2 ) . \lambda_0=c\lambda_*|\xi_0|=f'(0)(1+|\xi_0|^2). λ 0 = c λ ∗ ∣ ξ 0 ∣ = f ′ ( 0 ) ( 1 + ∣ ξ 0 ∣ 2 ) . This proves (81) for every k 0 ∈ K U k_0\in K_U k 0 ∈ K U
Remark 4.2. For fixed ( λ 0 , ξ 0 ) (\lambda_0,\xi_0) ( λ 0 , ξ 0 ) θ \theta θ ξ 0 ≠ 0 \xi_0\ne0 ξ 0 = 0 Ψ ( 0 ) = θ \Psi(0)=\theta Ψ ( 0 ) = θ Lemma 3.6 . If ξ 0 = 0 \xi_0=0 ξ 0 = 0 Y t = f ( Y ) Y_t=f(Y) Y t = f ( Y ) Y ( 0 ) = θ Y(0)=\theta Y ( 0 ) = θ Y n Y_n Y n C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2
The critical–subcritical decomposition
The parameter relation allows us to separate the region ∣ ξ ∣ < 1 |\xi|<1 ∣ ξ ∣ < 1
Proposition 4.3. For every U ∈ C U\in\mathcal{C} U ∈ C A , C ∈ C A,C\in\mathcal{C} A , C ∈ C A n ∈ C A_n\in\mathcal{C} A n ∈ C
U = A ⊕ C , A n ↗ A , A n ⊕ C ↗ U in C l o c 1 , 2 ( R × R N ) . (86) U=A\oplus C,\qquad A_n\nearrow A,\qquad A_n\oplus C\nearrow U\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N). \tag*{(86)} U = A ⊕ C , A n ↗ A , A n ⊕ C ↗ U in C loc 1 , 2 ( R × R N ) . (86) Every nonzero A n A_n A n
∣ ∇ ln A n ∣ ≤ λ ∗ r n , r n : = 1 − 1 n < 1 for n ≥ 3. (87) |\nabla\ln A_n|\le\lambda_*r_n,\qquad r_n:=1-\frac{1}{n}<1\qquad\text{for }n\ge3. \tag*{(87)} ∣∇ ln A n ∣ ≤ λ ∗ r n , r n := 1 − n 1 < 1 for n ≥ 3. (87) and has a finite associated linear field with measure supported in B ‾ r n \overline{B}_{r_n} B r n C C C
C ( t + s , x ) = D e 2 f ′ ( 0 ) s C ( t , x ) for every s ∈ R , ( t , x ) ∈ R × R N . (88) C(t+s,x)=D_{e^{2f'(0)s}}C(t,x)\qquad\text{for every }s\in\mathbb{R},\ (t,x)\in\mathbb{R}\times\mathbb{R}^N. \tag*{(88)} C ( t + s , x ) = D e 2 f ′ ( 0 ) s C ( t , x ) for every s ∈ R , ( t , x ) ∈ R × R N . (88) The zero solution is allowed for A A A C C C A n A_n A n
Proof. The case U = 0 U=0 U = 0 U ≠ 0 U\ne0 U = 0 n ≥ 3 n\ge3 n ≥ 3
χ n ( q ) = min { 1 , max { 0 , n ( 1 − q ) − 1 } } , 0 ≤ q ≤ 1 , A n = J − 1 ( χ n ( ∣ ξ ∣ ) ) . \chi_n(q)=\min\{1,\max\{0,n(1-q)-1\}\},\qquad0\le q\le1,\qquad A_n=\mathcal{J}^{-1}(\chi_n(|\xi|)). χ n ( q ) = min { 1 , max { 0 , n ( 1 − q ) − 1 }} , 0 ≤ q ≤ 1 , A n = J − 1 ( χ n ( ∣ ξ ∣ )) . The functions χ n \chi_n χ n n n n q ≥ r n q\ge r_n q ≥ r n q ≤ 1 − 2 / n q\le1-2/n q ≤ 1 − 2/ n
0 ≤ A n ≤ A n + 1 ≤ U . 0\le A_n\le A_{n+1}\le U. 0 ≤ A n ≤ A n + 1 ≤ U . For any h ∈ R N h \in\mathbb{R}^{N} h ∈ R N C ( K U ) C(K_{U}) C ( K U )
e − λ ∗ r n ∣ h ∣ χ n ( ∣ ξ ∣ ) ≤ e λ ∗ h ⋅ ξ χ n ( ∣ ξ ∣ ) ≤ e λ ∗ r n ∣ h ∣ χ n ( ∣ ξ ∣ ) . e^{-\lambda_{*}r_{n}|h|}\chi_{n}(|\xi|) \le e^{\lambda_{*}h\cdot\xi}\chi_{n}(|\xi|) \le e^{\lambda_{*}r_{n}|h|}\chi_{n}(|\xi|). e − λ ∗ r n ∣ h ∣ χ n ( ∣ ξ ∣ ) ≤ e λ ∗ h ⋅ ξ χ n ( ∣ ξ ∣ ) ≤ e λ ∗ r n ∣ h ∣ χ n ( ∣ ξ ∣ ) . Returning to entire solutions and then using the elementary dilation bounds yields
e − λ ∗ r n ∣ h ∣ A n ( t , x ) ≤ D e − λ ∗ r n ∣ h ∣ A n ( t , x ) ≤ A n ( t , x + h ) ≤ D e λ ∗ r n ∣ h ∣ A n ( t , x ) ≤ e λ ∗ r n ∣ h ∣ A n ( t , x ) . \begin{aligned}
e^{-\lambda_{*}r_{n}|h|}A_{n}(t,x) \le D e^{-\lambda_{*}r_{n}|h|}A_{n}(t,x) \le A_{n}(t,x+h) \\
&\le D e^{\lambda_{*}r_{n}|h|}A_{n}(t,x) \le e^{\lambda_{*}r_{n}|h|}A_{n}(t,x).
\end{aligned} e − λ ∗ r n ∣ h ∣ A n ( t , x ) ≤ D e − λ ∗ r n ∣ h ∣ A n ( t , x ) ≤ A n ( t , x + h ) ≤ D e λ ∗ r n ∣ h ∣ A n ( t , x ) ≤ e λ ∗ r n ∣ h ∣ A n ( t , x ) . If A n ≠ 0 A_{n} \ne0 A n = 0 (87) . This argument uses no amplitude differentiability. The finite-linear-lift theorem for a strict logarithmic gradient gap, Theorem 3.8 , gives the asserted measure supported in B ‾ r n \overline{B}_{r_{n}} B r n
Bounded monotone parabolic compactness gives
A n ↗ A in C l o c 1 , 2 , 0 ≤ A ≤ U , A ∈ C ∩ E U . A_{n} \nearrow A \text{ in } C_{\mathrm{loc}}^{1,2}, \qquad0 \le A \le U, \qquad A \in\mathcal{C} \cap E_{U}. A n ↗ A in C loc 1 , 2 , 0 ≤ A ≤ U , A ∈ C ∩ E U . This A A A A n A_{n} A n E E E A n A_{n} A n A A A C ∈ C C \in\mathcal{C} C ∈ C
C = U ⊖ A , U = A ⊕ C . C = U \ominus A, \qquad U = A \oplus C. C = U ⊖ A , U = A ⊕ C . Here ⊖ \ominus ⊖ 0 ≤ C ≤ U 0 \le C \le U 0 ≤ C ≤ U C ∈ E U C \in E_{U} C ∈ E U g = J ( C ) g = \mathcal{J}(C) g = J ( C )
0 ≤ g = 1 − J ( A ) ≤ 1 − χ n ( ∣ ξ ∣ ) for every n ≥ 3. 0 \le g = 1 - \mathcal{J}(A) \le1 - \chi_{n}(|\xi|) \qquad\text{for every } n \ge3. 0 ≤ g = 1 − J ( A ) ≤ 1 − χ n ( ∣ ξ ∣ ) for every n ≥ 3. At each k k k ∣ ξ ( k ) ∣ < 1 |\xi(k)| < 1 ∣ ξ ( k ) ∣ < 1 n n n
supp g ⊂ { k : ∣ ξ ( k ) ∣ = 1 } . \operatorname{supp} g \subset\{k : |\xi(k)| = 1\}. supp g ⊂ { k : ∣ ξ ( k ) ∣ = 1 } . By Theorem 4.1 , λ = 2 f ′ ( 0 ) \lambda= 2f'(0) λ = 2 f ′ ( 0 ) s ∈ R s \in\mathbb{R} s ∈ R
J ( T s , 0 C ) = e s λ g = e 2 f ′ ( 0 ) s g = J ( D e 2 f ′ ( 0 ) s C ) . \mathcal{J}(T_{s,0}C) = e^{s\lambda}g = e^{2f'(0)s}g = \mathcal{J}(D e^{2f'(0)s}C). J ( T s , 0 C ) = e s λ g = e 2 f ′ ( 0 ) s g = J ( D e 2 f ′ ( 0 ) s C ) . Injectivity of J \mathcal{J} J (88) . Finally, increasing continuity of superposition gives
A n ⊕ C ↗ A ⊕ C = U ; A_{n} \oplus C \nearrow A \oplus C = U; A n ⊕ C ↗ A ⊕ C = U ; bounded parabolic compactness upgrades this convergence to C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2 (86) . □
In particular, this construction does not require a prior critical projection, continuity of that projection, or preservation of the critical trace under a decreasing limit. It produces the critical covariant remainder directly from the spectral support of J ( U ⊖ A ) \mathcal{J}(U \ominus A) J ( U ⊖ A )
Construction and identification of critical entire solutions
We construct entire solutions from arbitrary finite positive measures on the critical sphere and prove that every critical component obtained in § 8 belongs to this family. This establishes Theorem 1.2 and identifies the part not covered by the strict gradient bound. Throughout this section the fixed reaction satisfies (2) , with f ′ ( 0 ) > 0 f'(0)>0 f ′ ( 0 ) > 0 (2) . We retain F 1 F_1 F 1 (46) , write S t S_t S t
F [ v ] = v t − Δ v − f ( v ) , M f = 1 2 max 0 ≤ u ≤ 1 ∣ f ′ ′ ( u ) ∣ . \mathcal{F}[v]=v_t-\Delta v-f(v),\qquad M_f=\frac{1}{2}\max_{0\leq u\leq1}\lvert f''(u)\rvert. F [ v ] = v t − Δ v − f ( v ) , M f = 2 1 0 ≤ u ≤ 1 max ∣ f ′′ ( u )∣ . Concavity gives f ′ ( u ) ≤ f ′ ( 0 ) f'(u)\leq f'(0) f ′ ( u ) ≤ f ′ ( 0 ) u ≥ 0 u\geq0 u ≥ 0 0 ≤ F 1 ( u ) ≤ M f u 2 0\leq F_1(u)\leq M_f u^2 0 ≤ F 1 ( u ) ≤ M f u 2 0 ≤ u ≤ 1 0\leq u\leq1 0 ≤ u ≤ 1
For a finite positive Borel measure μ \mu μ S N − 1 \mathbb{S}^{N-1} S N − 1 L μ L_\mu L μ z μ z_\mu z μ (12) . We first construct a radially symmetric solution. It provides a uniform upper bound for the subsequent construction with arbitrary finite Borel measures on S N − 1 \mathbb{S}^{N-1} S N − 1
Fix one translate of the increasing critical wave, satisfying
ϕ ∗ ′ ′ − c ∗ ϕ ∗ ′ + f ( ϕ ∗ ) = 0 , ϕ ∗ ( − ∞ ) = 0 , ϕ ∗ ( + ∞ ) = 1. \phi_*''-c_*\phi_*'+f(\phi_*)=0,\qquad\phi_*(-\infty)=0,\qquad\phi_*(+\infty)=1. ϕ ∗ ′′ − c ∗ ϕ ∗ ′ + f ( ϕ ∗ ) = 0 , ϕ ∗ ( − ∞ ) = 0 , ϕ ∗ ( + ∞ ) = 1. The wave estimates in Subsection 3.2, together with (6) , give
ϕ ∗ ( s ) ∼ ( − s ) e λ ∗ s ( s → − ∞ ) , ∣ ϕ ∗ ( j ) ( s ) ∣ ≤ C ( 1 + ∣ s ∣ ) e λ ∗ s for s ≤ − 1 , j = 0 , 1 , 2. (89) \phi_*(s)\sim(-s)e^{\lambda_*s}\quad(s\to-\infty),\qquad\lvert\phi_*^{(j)}(s)\rvert\leq C(1+\lvert s\rvert)e^{\lambda_*s}\quad\text{for }s\leq-1,\ j=0,1,2. \tag*{(89)} ϕ ∗ ( s ) ∼ ( − s ) e λ ∗ s ( s → − ∞ ) , ∣ ϕ ∗ ( j ) ( s )∣ ≤ C ( 1 + ∣ s ∣) e λ ∗ s for s ≤ − 1 , j = 0 , 1 , 2. (89) We also need ϕ ∗ ′ ′ − λ ∗ ϕ ∗ ′ ≤ 0 \phi_*''-\lambda_*\phi_*'\leq0 ϕ ∗ ′′ − λ ∗ ϕ ∗ ′ ≤ 0 h = ϕ ∗ ′ ′ − λ ∗ ϕ ∗ ′ = λ ∗ ϕ ∗ ′ − f ( ϕ ∗ ) h=\phi_*''-\lambda_*\phi_*'=\lambda_*\phi_*'-f(\phi_*) h = ϕ ∗ ′′ − λ ∗ ϕ ∗ ′ = λ ∗ ϕ ∗ ′ − f ( ϕ ∗ )
h ′ − λ ∗ h = ( f ′ ( 0 ) − f ′ ( ϕ ∗ ) ) ϕ ∗ ′ ≥ 0. h'-\lambda_*h=(f'(0)-f'(\phi_*))\phi_*'\geq0. h ′ − λ ∗ h = ( f ′ ( 0 ) − f ′ ( ϕ ∗ )) ϕ ∗ ′ ≥ 0. The boundedness of the profile and its first two derivatives gives e − λ ∗ s h ( s ) → 0 e^{-\lambda_*s}h(s)\to0 e − λ ∗ s h ( s ) → 0 + ∞ +\infty + ∞ ( e − λ ∗ s h ) ′ ≥ 0 (e^{-\lambda_*s}h)'\geq0 ( e − λ ∗ s h ) ′ ≥ 0 h ≤ 0 h\leq0 h ≤ 0 R \mathbb{R} R
g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) for q > 0 , g ( 0 ) = 0 , g(q)=\phi_*(\lambda_*^{-1}\ln q)\quad\text{for }q>0,\qquad g(0)=0, g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) for q > 0 , g ( 0 ) = 0 , we have g ′ ′ ( q ) = λ ∗ − 2 q − 2 h ( λ ∗ − 1 ln q ) ≤ 0 g''(q)=\lambda_*^{-2}q^{-2}h(\lambda_*^{-1}\ln q)\leq0 g ′′ ( q ) = λ ∗ − 2 q − 2 h ( λ ∗ − 1 ln q ) ≤ 0 g g g [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
The radially symmetric solution
We construct the radially symmetric solution that will bound the solutions generated by general critical-sphere measures. The main step is to build a supersolution that decays on compact sets in the remote past and dominates the critical-profile subsolution.
Throughout this subsection N ≥ 2 N\geq2 N ≥ 2 σ \sigma σ S N − 1 \mathbb{S}^{N-1} S N − 1
∫ S N − 1 d σ ( ν ) = σ ( S N − 1 ) = 1. \int_{\mathbb{S}^{N-1}}d\sigma(\nu)=\sigma(\mathbb{S}^{N-1})=1. ∫ S N − 1 d σ ( ν ) = σ ( S N − 1 ) = 1. The following theorem constructs the radially symmetric entire solution that will provide an upper bound for solutions associated with general measures.
Theorem 5.1. There exists a radially symmetric entire solution u σ = U r a d u_\sigma=U^{\mathrm{rad}} u σ = U rad
0 < u σ < 1 , ∂ t u σ > 0 , ϕ ∗ ( z σ ) ≤ u σ , 0<u_\sigma<1,\qquad\partial_tu_\sigma>0,\qquad\phi_*(z_\sigma)\leq u_\sigma, 0 < u σ < 1 , ∂ t u σ > 0 , ϕ ∗ ( z σ ) ≤ u σ , and, for every M > 0 , M>0, M > 0 ,
lim t → − ∞ sup ∣ x ∣ ≤ M u σ ( t , x ) = 0. \lim_{t\to-\infty}\sup_{\lvert x\rvert\leq M}u_\sigma(t,x)=0. t → − ∞ lim ∣ x ∣ ≤ M sup u σ ( t , x ) = 0. It is minimal above ϕ ∗ ( z σ ) \phi_*(z_\sigma) ϕ ∗ ( z σ ) W ∈ E W\in\mathcal{E} W ∈ E W ≥ ϕ ∗ ( z σ ) W\geq\phi_*(z_\sigma) W ≥ ϕ ∗ ( z σ ) W ≥ u σ W\geq u_\sigma W ≥ u σ b > 0 b>0 b > 0 ϕ ∗ ( z b σ ) \phi_*(zb\sigma) ϕ ∗ ( z bσ )
u b σ ( t , x ) = u σ ( t + ln b 2 f ′ ( 0 ) , x ) . u_{b\sigma}(t,x)=u_\sigma\left(t+\frac{\ln b}{2f'(0)},x\right). u bσ ( t , x ) = u σ ( t + 2 f ′ ( 0 ) ln b , x ) . Proof. Set F [ v ] : = v t − Δ v − f ( v ) \mathcal{F}[v] := v_t - \Delta v - f(v) F [ v ] := v t − Δ v − f ( v ) L σ L_\sigma L σ z σ z_\sigma z σ (12) . Since the surface area measure σ \sigma σ L σ L_\sigma L σ L σ ( x ) = L σ ( ∣ x ∣ e 1 ) L_\sigma(x) = L_\sigma(|x|e_1) L σ ( x ) = L σ ( ∣ x ∣ e 1 ) e 1 = ( 1 , 0 , ⋯ , 0 ) e_1 = (1,0,\cdots,0) e 1 = ( 1 , 0 , ⋯ , 0 ) z σ ( t , r ) = z σ ( t , r e 1 ) z_\sigma(t,r) = z_\sigma(t,re_1) z σ ( t , r ) = z σ ( t , r e 1 ) r ≥ 0 r \ge0 r ≥ 0 L σ L_\sigma L σ Δ L σ = f ′ ( 0 ) L σ \Delta L_\sigma= f'(0)L_\sigma Δ L σ = f ′ ( 0 ) L σ
∂ r r z σ + λ ∗ ( ∂ r z σ ) 2 + N − 1 r ∂ r z σ = λ ∗ for t ∈ R , r > 0. (90) \partial_{rr}z_\sigma+\lambda_*(\partial_r z_\sigma)^2+\frac{N-1}{r}\partial_r z_\sigma=\lambda_*
\quad\text{for } t\in\mathbb{R},\ r>0. \tag*{(90)} ∂ r r z σ + λ ∗ ( ∂ r z σ ) 2 + r N − 1 ∂ r z σ = λ ∗ for t ∈ R , r > 0. (90) Differentiation under the integral gives
∂ r z σ ( t , r ) = ∫ S N − 1 ν 1 e λ ∗ r ν 1 σ ( d ν ) L σ ( r e 1 ) , \partial_r z_\sigma(t,r)=\frac{\int_{\mathbb{S}^{N-1}}\nu_1e^{\lambda_*r\nu_1}\,\sigma(d\nu)}{L_\sigma(re_1)}, ∂ r z σ ( t , r ) = L σ ( r e 1 ) ∫ S N − 1 ν 1 e λ ∗ r ν 1 σ ( d ν ) , ∂ r r z σ ( t , r ) = λ ∗ ∫ S N − 1 × S N − 1 ( ν 1 − η 1 ) 2 e λ ∗ r ( ν 1 + η 1 ) σ ( d ν ) σ ( d η ) 2 L σ ( r e 1 ) 2 . \partial_{rr}z_\sigma(t,r)=\frac{\lambda_*\int_{\mathbb{S}^{N-1}\times\mathbb{S}^{N-1}}(\nu_1-\eta_1)^2e^{\lambda_*r(\nu_1+\eta_1)}\,\sigma(d\nu)\sigma(d\eta)}{2L_\sigma(re_1)^2}. ∂ r r z σ ( t , r ) = 2 L σ ( r e 1 ) 2 λ ∗ ∫ S N − 1 × S N − 1 ( ν 1 − η 1 ) 2 e λ ∗ r ( ν 1 + η 1 ) σ ( d ν ) σ ( d η ) . Clearly,
0 < ∂ r z σ ( t , r ) < 1 , ∂ r r z σ ( t , r ) ≥ 0 for t ∈ R , r > 0. (91) 0<\partial_r z_\sigma(t,r)<1,\qquad\partial_{rr}z_\sigma(t,r)\ge0
\quad\text{for } t\in\mathbb{R},\ r>0. \tag*{(91)} 0 < ∂ r z σ ( t , r ) < 1 , ∂ r r z σ ( t , r ) ≥ 0 for t ∈ R , r > 0. (91) Here are the elementary bounds that will be used below:
c ( 1 + r ) − N − 1 2 e λ ∗ r ≤ L σ ( r e 1 ) ≤ C ( 1 + r ) − N − 1 2 e λ ∗ r for r ≥ 0 , (92) c(1+r)^{-\frac{N-1}{2}}e^{\lambda_*r}\le L_\sigma(re_1)\le C(1+r)^{-\frac{N-1}{2}}e^{\lambda_*r}
\quad\text{for } r\ge0, \tag*{(92)} c ( 1 + r ) − 2 N − 1 e λ ∗ r ≤ L σ ( r e 1 ) ≤ C ( 1 + r ) − 2 N − 1 e λ ∗ r for r ≥ 0 , (92) and
0 ≤ 1 − ∂ r z σ ( t , r ) ≤ C r for t ∈ R , r ≥ 1. (93) 0\le1-\partial_r z_\sigma(t,r)\le\frac{C}{r}
\quad\text{for } t\in\mathbb{R},\ r\ge1. \tag*{(93)} 0 ≤ 1 − ∂ r z σ ( t , r ) ≤ r C for t ∈ R , r ≥ 1. (93) Indeed, the integral representation of the modified Bessel function of the first kind gives
L σ ( r e 1 ) = Γ ( N 2 ) ( 2 λ ∗ r ) N 2 − 1 I N 2 − 1 ( λ ∗ r ) for r > 0. L_\sigma(re_1)=\Gamma\left(\frac{N}{2}\right)\left(\frac{2}{\lambda_*r}\right)^{\frac{N}{2}-1}I_{\frac{N}{2}-1}(\lambda_*r)
\quad\text{for } r>0. L σ ( r e 1 ) = Γ ( 2 N ) ( λ ∗ r 2 ) 2 N − 1 I 2 N − 1 ( λ ∗ r ) for r > 0. Its differentiation formula and expansion yield
L σ ( r e 1 ) = 2 N 2 − 1 Γ ( N / 2 ) 2 π ( λ ∗ r ) − N − 1 2 e λ ∗ r ( 1 + O ( r − 1 ) ) , L_\sigma(re_1)=\frac{2^{\frac{N}{2}-1}\Gamma(N/2)}{\sqrt{2\pi}}(\lambda_*r)^{-\frac{N-1}{2}}e^{\lambda_*r}(1+O(r^{-1})), L σ ( r e 1 ) = 2 π 2 2 N − 1 Γ ( N /2 ) ( λ ∗ r ) − 2 N − 1 e λ ∗ r ( 1 + O ( r − 1 )) , ∂ r z σ ( t , r ) = I N 2 ( λ ∗ r ) I N 2 − 1 ( λ ∗ r ) = 1 − N − 1 2 λ ∗ r + O ( r − 2 ) , as r → ∞ . \partial_r z_\sigma(t,r)=\frac{I_{\frac{N}{2}}(\lambda_*r)}{I_{\frac{N}{2}-1}(\lambda_*r)}
=1-\frac{N-1}{2\lambda_*r}+O(r^{-2}),
\qquad\text{as } r\to\infty. ∂ r z σ ( t , r ) = I 2 N − 1 ( λ ∗ r ) I 2 N ( λ ∗ r ) = 1 − 2 λ ∗ r N − 1 + O ( r − 2 ) , as r → ∞. Together with (91) and positivity and continuity on bounded intervals, these imply (92) and (93) .
For t = − T < 0 t=-T<0 t = − T < 0 R ( t ) > 0 R(t)>0 R ( t ) > 0
z σ ( t , R ( t ) ) = 0. (94) z_\sigma(t,R(t))=0. \tag*{(94)} z σ ( t , R ( t )) = 0. (94) Existence and uniqueness follow from z σ ( t , 0 ) = c ∗ t < 0 z_\sigma(t,0)=c_*t<0 z σ ( t , 0 ) = c ∗ t < 0 ∂ r z σ ( t , r ) > 0 \partial_rz_\sigma(t,r)>0 ∂ r z σ ( t , r ) > 0 r > 0 r>0 r > 0 z σ ( t , r ) → ∞ z_\sigma(t,r)\to\infty z σ ( t , r ) → ∞ r → ∞ r\to\infty r → ∞ ∂ t z σ = c ∗ \partial_tz_\sigma=c_* ∂ t z σ = c ∗ (92) , (93) , and (94) imply
R ( − T ) = c ∗ T + N − 1 c ∗ ln T + O ( 1 ) , R ′ ( t ) = − c ∗ ∂ r z σ ( t , R ( t ) ) , c ∗ ≤ − R ′ ( t ) ≤ C (95) R(-T)=c_*T+\frac{N-1}{c_*}\ln T+O(1),\qquad
R'(t)=-\frac{c_*}{\partial_rz_\sigma(t,R(t))},\qquad
c_*\le-R'(t)\le C \tag*{(95)} R ( − T ) = c ∗ T + c ∗ N − 1 ln T + O ( 1 ) , R ′ ( t ) = − ∂ r z σ ( t , R ( t )) c ∗ , c ∗ ≤ − R ′ ( t ) ≤ C (95) for all sufficiently large T T T
Define
u ( t , x ) : = ϕ ∗ ( z σ ( t , x ) ) . u(t,x):=\phi_*(z_\sigma(t,x)). u ( t , x ) := ϕ ∗ ( z σ ( t , x )) . The function z σ z_\sigma z σ x = 0 x=0 x = 0 H 0 = e 2 f ′ ( 0 ) t L σ ( x ) H_0=e^{2f'(0)t}L_\sigma(x) H 0 = e 2 f ′ ( 0 ) t L σ ( x ) ( H 0 ) t − Δ H 0 = f ′ ( 0 ) H 0 (H_0)_t-\Delta H_0=f'(0)H_0 ( H 0 ) t − Δ H 0 = f ′ ( 0 ) H 0 ∣ ∇ H 0 ∣ ≤ λ ∗ H 0 |\nabla H_0|\leq\lambda_*H_0 ∣∇ H 0 ∣ ≤ λ ∗ H 0
∂ t z σ − Δ z σ = λ ∗ ( 1 + ∣ ∇ z σ ∣ 2 ) , ∣ ∇ z σ ∣ ≤ 1. \partial_tz_\sigma-\Delta z_\sigma=\lambda_*(1+|\nabla z_\sigma|^2),\qquad|\nabla z_\sigma|\leq1. ∂ t z σ − Δ z σ = λ ∗ ( 1 + ∣∇ z σ ∣ 2 ) , ∣∇ z σ ∣ ≤ 1. The wave equation therefore yields
F [ u ] = ( 1 − ∣ ∇ z σ ∣ 2 ) ( ϕ ∗ ′ ′ ( z σ ) − λ ∗ ϕ ∗ ′ ( z σ ) ) ≤ 0. (96) \mathcal{F}[u]=(1-|\nabla z_\sigma|^2)(\phi_*''(z_\sigma)-\lambda_*\phi_*'(z_\sigma))\leq0. \tag*{(96)} F [ u ] = ( 1 − ∣∇ z σ ∣ 2 ) ( ϕ ∗ ′′ ( z σ ) − λ ∗ ϕ ∗ ′ ( z σ )) ≤ 0. (96) Thus u u u
We now construct a bounded supersolution for all sufficiently negative times. Put
d ( t , x ) : = ∣ x ∣ − R ( t ) , L ( t ) : = R ( t ) ( ln R ( t ) ) 2 , η ( t ) : = D ln R ( t ) , d(t,x):=|x|-R(t),\qquad L(t):=\frac{R(t)}{(\ln R(t))^2},\qquad\eta(t):=\frac{D}{\ln R(t)}, d ( t , x ) := ∣ x ∣ − R ( t ) , L ( t ) := ( ln R ( t ) ) 2 R ( t ) , η ( t ) := ln R ( t ) D , where D > 0 D>0 D > 0 d d d x ≠ 0 x\neq0 x = 0 d ≥ − 3 L ( t ) d\geq-3L(t) d ≥ − 3 L ( t ) R ( t ) R(t) R ( t ) ∣ x ∣ ≥ R ( t ) / 2 |x|\geq R(t)/2 ∣ x ∣ ≥ R ( t ) /2
∣ ∇ d ∣ = 1 , d t = − R ′ ( t ) , Δ d = N − 1 R ( t ) + d . |\nabla d|=1,\qquad d_t=-R'(t),\qquad\Delta d=\frac{N-1}{R(t)+d}. ∣∇ d ∣ = 1 , d t = − R ′ ( t ) , Δ d = R ( t ) + d N − 1 . At d = 0 d=0 d = 0 (90) gives the exact inequality
− R ′ ( t ) − N − 1 R ( t ) − c ∗ = c ∗ ∂ r z σ ( t , R ( t ) ) − N − 1 R ( t ) − c ∗ = λ ∗ ( 1 − ∂ r z σ ( t , R ( t ) ) ) 2 + ∂ r r z σ ( t , R ( t ) ) ∂ r z σ ( t , R ( t ) ) ≥ 0. \begin{aligned}
-R'(t)-\frac{N-1}{R(t)}-c_*=\frac{c_*}{\partial_rz_\sigma(t,R(t))}-\frac{N-1}{R(t)}-c_* \\
&=\frac{\lambda_*(1-\partial_rz_\sigma(t,R(t)))^2+\partial_{rr}z_\sigma(t,R(t))}{\partial_rz_\sigma(t,R(t))}\geq0.
\end{aligned} − R ′ ( t ) − R ( t ) N − 1 − c ∗ = ∂ r z σ ( t , R ( t )) c ∗ − R ( t ) N − 1 − c ∗ = ∂ r z σ ( t , R ( t )) λ ∗ ( 1 − ∂ r z σ ( t , R ( t )) ) 2 + ∂ r r z σ ( t , R ( t )) ≥ 0. For − 3 L ( t ) ≤ d ≤ 0 -3L(t)\leq d\leq0 − 3 L ( t ) ≤ d ≤ 0 d t = − R ′ ( t ) d_t=-R'(t) d t = − R ′ ( t ) Δ d = ( N − 1 ) / ( R ( t ) + d ) \Delta d=(N-1)/(R(t)+d) Δ d = ( N − 1 ) / ( R ( t ) + d )
d t − Δ d − c ∗ = ( − R ′ ( t ) − N − 1 R ( t ) − c ∗ ) + ( N − 1 ) ( 1 R ( t ) − 1 R ( t ) + d ) ≥ ( N − 1 ) d R ( t ) ( R ( t ) + d ) ≥ − C L ( t ) R ( t ) 2 , \begin{aligned}
d_t-\Delta d-c_*=\left(-R'(t)-\frac{N-1}{R(t)}-c_*\right)+(N-1)\left(\frac{1}{R(t)}-\frac{1}{R(t)+d}\right) \\
&\geq\frac{(N-1)d}{R(t)(R(t)+d)}\geq-\frac{CL(t)}{R(t)^2},
\end{aligned} d t − Δ d − c ∗ = ( − R ′ ( t ) − R ( t ) N − 1 − c ∗ ) + ( N − 1 ) ( R ( t ) 1 − R ( t ) + d 1 ) ≥ R ( t ) ( R ( t ) + d ) ( N − 1 ) d ≥ − R ( t ) 2 C L ( t ) , where the last inequality uses ∣ d ∣ ≤ 3 L ( t ) |d|\leq3L(t) ∣ d ∣ ≤ 3 L ( t ) R ( t ) + d ≥ R ( t ) / 2 R(t)+d\geq R(t)/2 R ( t ) + d ≥ R ( t ) /2 t t t d ≥ 0 d\geq0 d ≥ 0 d t − Δ d − c ∗ ≥ 0 d_t-\Delta d-c_*\geq0 d t − Δ d − c ∗ ≥ 0
η ′ ( t ) = − D R ′ ( t ) R ( t ) ( ln R ( t ) ) 2 ≥ c ∗ D R ( t ) ( ln R ( t ) ) 2 . \eta'(t)=\frac{-DR'(t)}{R(t)(\ln R(t))^2}\geq\frac{c_*D}{R(t)(\ln R(t))^2}. η ′ ( t ) = R ( t ) ( ln R ( t ) ) 2 − D R ′ ( t ) ≥ R ( t ) ( ln R ( t ) ) 2 c ∗ D . Since L ( t ) / R ( t ) 2 = 1 / ( R ( t ) ( ln R ( t ) ) 2 ) L(t)/R(t)^2=1/(R(t)(\ln R(t))^2) L ( t ) / R ( t ) 2 = 1/ ( R ( t ) ( ln R ( t ) ) 2 )
v ( t , x ) : = ϕ ∗ ( d ( t , x ) + η ( t ) ) v(t,x):=\phi_*(d(t,x)+\eta(t)) v ( t , x ) := ϕ ∗ ( d ( t , x ) + η ( t )) satisfies
F [ v ] = ϕ ∗ ′ ( d + η ( t ) ) ( d t + η ′ ( t ) − Δ d − c ∗ ) ≥ 0 for d ≥ − 3 L ( t ) , (97) \mathcal{F}[v]=\phi_*'(d+\eta(t))(d_t+\eta'(t)-\Delta d-c_*)\geq0\qquad\text{for }d\geq-3L(t), \tag*{(97)} F [ v ] = ϕ ∗ ′ ( d + η ( t )) ( d t + η ′ ( t ) − Δ d − c ∗ ) ≥ 0 for d ≥ − 3 L ( t ) , (97) by taking D D D
The supersolution inequality above is only established for d ≥ − 3 L ( t ) d\geq-3L(t) d ≥ − 3 L ( t ) d = ∣ x ∣ − R ( t ) d=|x|-R(t) d = ∣ x ∣ − R ( t ) v v v 0 ≤ χ ≤ 1 0\leq\chi\leq1 0 ≤ χ ≤ 1 χ ( s ) = 0 \chi(s)=0 χ ( s ) = 0 s ≤ − 2 s\leq-2 s ≤ − 2 χ ( s ) = 1 \chi(s)=1 χ ( s ) = 1 s ≥ − 1 s\geq-1 s ≥ − 1
Ψ ( t , x ) : = χ ( d ( t , x ) / L ( t ) ) ϕ ∗ ( d ( t , x ) + η ( t ) ) . \Psi(t,x):=\chi(d(t,x)/L(t))\phi_*(d(t,x)+\eta(t)). Ψ ( t , x ) := χ ( d ( t , x ) / L ( t )) ϕ ∗ ( d ( t , x ) + η ( t )) . It agrees with v v v d ≥ − L ( t ) d\geq-L(t) d ≥ − L ( t ) d ≤ − 2 L ( t ) d\leq-2L(t) d ≤ − 2 L ( t )
On the strip − 2 L ( t ) ≤ d ≤ − L ( t ) -2L(t) \le d \le-L(t) − 2 L ( t ) ≤ d ≤ − L ( t ) d t = O ( 1 ) d_t = O(1) d t = O ( 1 ) Δ d = O ( R ( t ) − 1 ) \Delta d = O(R(t)^{-1}) Δ d = O ( R ( t ) − 1 ) L ′ ( t ) = O ( ( ln R ( t ) ) − 2 ) L'(t) = O((\ln R(t))^{-2}) L ′ ( t ) = O (( ln R ( t ) ) − 2 )
∣ ∇ [ χ ( d L ( t ) ) ] ∣ ≤ C L ( t ) , ∣ ( ∂ t − Δ ) [ χ ( d L ( t ) ) ] ∣ ≤ C L ( t ) . \left|\nabla\left[\chi\left(\frac{d}{L(t)}\right)\right]\right| \le\frac{C}{L(t)}, \qquad\left|(\partial_t-\Delta)\left[\chi\left(\frac{d}{L(t)}\right)\right]\right| \le\frac{C}{L(t)}. ∇ [ χ ( L ( t ) d ) ] ≤ L ( t ) C , ( ∂ t − Δ ) [ χ ( L ( t ) d ) ] ≤ L ( t ) C . The product identity is
F [ Ψ ] = χ ( d L ( t ) ) F [ v ] + v ( ∂ t − Δ ) [ χ ( d L ( t ) ) ] − 2 ∇ [ χ ( d L ( t ) ) ] ⋅ ∇ v + χ ( d L ( t ) ) f ( v ) − f ( χ ( d L ( t ) ) v ) . \begin{aligned}
\mathcal{F}[\Psi] ={}& \chi\left(\frac{d}{L(t)}\right)\mathcal{F}[v] + v(\partial_t-\Delta)\left[\chi\left(\frac{d}{L(t)}\right)\right] \\
&-2\nabla\left[\chi\left(\frac{d}{L(t)}\right)\right]\cdot\nabla v+\chi\left(\frac{d}{L(t)}\right)f(v)-f\left(\chi\left(\frac{d}{L(t)}\right)v\right).
\end{aligned} F [ Ψ ] = χ ( L ( t ) d ) F [ v ] + v ( ∂ t − Δ ) [ χ ( L ( t ) d ) ] − 2∇ [ χ ( L ( t ) d ) ] ⋅ ∇ v + χ ( L ( t ) d ) f ( v ) − f ( χ ( L ( t ) d ) v ) . The first term is nonnegative by (97) . The last two terms satisfy
χ ( d L ( t ) ) f ( v ) − f ( χ ( d L ( t ) ) v ) = − χ ( d L ( t ) ) F 1 ( v ) + F 1 ( χ ( d L ( t ) ) v ) ≥ − M f v 2 . \chi\left(\frac{d}{L(t)}\right)f(v)-f\left(\chi\left(\frac{d}{L(t)}\right)v\right)=-\chi\left(\frac{d}{L(t)}\right)F_1(v)+F_1\left(\chi\left(\frac{d}{L(t)}\right)v\right)\ge-M_fv^2. χ ( L ( t ) d ) f ( v ) − f ( χ ( L ( t ) d ) v ) = − χ ( L ( t ) d ) F 1 ( v ) + F 1 ( χ ( L ( t ) d ) v ) ≥ − M f v 2 . Lemma 3.6 and (6) , together with η ( t ) → 0 \eta(t)\to0 η ( t ) → 0 L ( t ) → ∞ L(t)\to\infty L ( t ) → ∞ t → − ∞ t\to-\infty t → − ∞ v v v ∣ ∇ v ∣ |\nabla v| ∣∇ v ∣ C ( 1 + L ( t ) ) e λ ∗ d C(1+L(t))e^{\lambda_*d} C ( 1 + L ( t )) e λ ∗ d
F [ Ψ ] ≥ − C ( 1 + L ( t ) ) e λ ∗ d for − 2 L ( t ) ≤ d ≤ − L ( t ) . (98) \mathcal{F}[\Psi]\ge-C(1+L(t))e^{\lambda_*d}\qquad\text{for }-2L(t)\le d\le-L(t). \tag*{(98)} F [ Ψ ] ≥ − C ( 1 + L ( t )) e λ ∗ d for − 2 L ( t ) ≤ d ≤ − L ( t ) . (98) Outside this strip, F [ Ψ ] ≥ 0 \mathcal{F}[\Psi]\ge0 F [ Ψ ] ≥ 0
The cutoff introduces an error on − 2 L ( t ) ≤ d ( t , x ) ≤ − L ( t ) -2L(t)\le d(t,x)\le-L(t) − 2 L ( t ) ≤ d ( t , x ) ≤ − L ( t ) (98) . To compensate for this error, we add a correction term H H H Ψ \Psi Ψ H H H p > 0 p>0 p > 0
K ( t , x ) : = ∫ 0 1 e f ′ ( 0 ) ( 1 + ℓ 2 ) t L σ ( ℓ x ) d ℓ , H ( t , x ) : = R ( t ) − p K ( t , x ) . K(t,x):=\int_0^1 e^{f'(0)(1+\ell^2)t}L_\sigma(\ell x)\,d\ell,\qquad H(t,x):=R(t)^{-p}K(t,x). K ( t , x ) := ∫ 0 1 e f ′ ( 0 ) ( 1 + ℓ 2 ) t L σ ( ℓ x ) d ℓ , H ( t , x ) := R ( t ) − p K ( t , x ) . For each ℓ ∈ [ 0 , 1 ] \ell\in[0,1] ℓ ∈ [ 0 , 1 ] e f ′ ( 0 ) ( 1 + ℓ 2 ) t L σ ( ℓ x ) e^{f'(0)(1+\ell^2)t}L_\sigma(\ell x) e f ′ ( 0 ) ( 1 + ℓ 2 ) t L σ ( ℓ x )
( ∂ t − Δ − f ′ ( 0 ) ) [ e f ′ ( 0 ) ( 1 + ℓ 2 ) t L σ ( ℓ x ) ] = 0 , (\partial_t-\Delta-f'(0))\left[e^{f'(0)(1+\ell^2)t}L_\sigma(\ell x)\right]=0, ( ∂ t − Δ − f ′ ( 0 )) [ e f ′ ( 0 ) ( 1 + ℓ 2 ) t L σ ( ℓ x ) ] = 0 , because Δ L σ ( ℓ x ) = f ′ ( 0 ) ℓ 2 L σ ( ℓ x ) \Delta L_\sigma(\ell x)=f'(0)\ell^2L_\sigma(\ell x) Δ L σ ( ℓ x ) = f ′ ( 0 ) ℓ 2 L σ ( ℓ x ) K t − Δ K = f ′ ( 0 ) K K_t-\Delta K=f'(0)K K t − Δ K = f ′ ( 0 ) K K K K R ( t ) − p R(t)^{-p} R ( t ) − p
H t − Δ H − f ′ ( 0 ) H = β ( t ) H , β ( t ) : = − p R ′ ( t ) R ( t ) , c T ≤ β ( − T ) ≤ C T . (99) H_t-\Delta H-f'(0)H=\beta(t)H,\qquad\beta(t):=-p\frac{R'(t)}{R(t)},\qquad\frac{c}{T}\le\beta(-T)\le\frac{C}{T}. \tag*{(99)} H t − Δ H − f ′ ( 0 ) H = β ( t ) H , β ( t ) := − p R ( t ) R ′ ( t ) , T c ≤ β ( − T ) ≤ T C . (99) The correction term H H H M > 0 M>0 M > 0 ∣ x ∣ ≤ M |x|\le M ∣ x ∣ ≤ M L σ ( ℓ x ) ≤ e λ ∗ M L_\sigma(\ell x)\le e^{\lambda_*M} L σ ( ℓ x ) ≤ e λ ∗ M
sup ∣ x ∣ ≤ M K ( − T , x ) ≤ C M e − f ′ ( 0 ) T ∫ 0 1 e − f ′ ( 0 ) T ℓ 2 d ℓ ≤ C M T − 1 / 2 e − f ′ ( 0 ) T . (100) \sup_{|x|\le M}K(-T,x)\le C_Me^{-f'(0)T}\int_0^1e^{-f'(0)T\ell^2}\,d\ell\le C_MT^{-1/2}e^{-f'(0)T}. \tag*{(100)} ∣ x ∣ ≤ M sup K ( − T , x ) ≤ C M e − f ′ ( 0 ) T ∫ 0 1 e − f ′ ( 0 ) T ℓ 2 d ℓ ≤ C M T − 1/2 e − f ′ ( 0 ) T . (100) We next show that the correction term H H H d ( − T , x ) ≤ − L ( − T ) d(-T,x)\le-L(-T) d ( − T , x ) ≤ − L ( − T ) (95) , for a fixed c 0 > 0 c_0>0 c 0 > 0 T T T
0 ≤ r : = ∣ x ∣ ≤ R ( − T ) − L ( − T ) ≤ c ∗ T − c 0 T ( ln T ) 2 . (101) 0\le r:=|x|\le R(-T)-L(-T)\le c_*T-\frac{c_0T}{(\ln T)^2}. \tag*{(101)} 0 ≤ r := ∣ x ∣ ≤ R ( − T ) − L ( − T ) ≤ c ∗ T − ( ln T ) 2 c 0 T . (101) In particular, r ≤ c ∗ T − T r \le c_*T-\sqrt{T} r ≤ c ∗ T − T ℓ 0 = r / ( c ∗ T ) \ell_0=r/(c_*T) ℓ 0 = r / ( c ∗ T ) [ ℓ 0 , ℓ 0 + 1 / ( 2 λ ∗ T ) ] [\ell_0,\ell_0+1/(2\lambda_*\sqrt{T})] [ ℓ 0 , ℓ 0 + 1/ ( 2 λ ∗ T )] [ 0 , 1 ] [0,1] [ 0 , 1 ] (92) gives
K ( − T , x ) ≥ c e − f ′ ( 0 ) T + r 2 / ( 4 T ) ∫ 0 1 ( 1 + λ ∗ ℓ r ) − N − 1 2 e − f ′ ( 0 ) T ( ℓ − ℓ 0 ) 2 d ℓ ≥ c T − 1 / 2 ( 1 + r 2 / T ) − N − 1 2 e − f ′ ( 0 ) T + r 2 / ( 4 T ) . K(-T,x)\ge ce^{-f'(0)T+r^2/(4T)}\int_0^1(1+\lambda_*\ell r)^{-\frac{N-1}{2}}e^{-f'(0)T(\ell-\ell_0)^2}\,d\ell
\ge cT^{-1/2}(1+r^2/T)^{-\frac{N-1}{2}}e^{-f'(0)T+r^2/(4T)}. K ( − T , x ) ≥ c e − f ′ ( 0 ) T + r 2 / ( 4 T ) ∫ 0 1 ( 1 + λ ∗ ℓ r ) − 2 N − 1 e − f ′ ( 0 ) T ( ℓ − ℓ 0 ) 2 d ℓ ≥ c T − 1/2 ( 1 + r 2 / T ) − 2 N − 1 e − f ′ ( 0 ) T + r 2 / ( 4 T ) . For ℓ ∈ [ ℓ 0 , ℓ 0 + 1 / ( 2 λ ∗ T ) ] \ell\in[\ell_0,\ell_0+1/(2\lambda_*\sqrt{T})] ℓ ∈ [ ℓ 0 , ℓ 0 + 1/ ( 2 λ ∗ T )] f ′ ( 0 ) T ( ℓ − ℓ 0 ) 2 ≤ 1 / 4 f'(0)T(\ell-\ell_0)^2\le1/4 f ′ ( 0 ) T ( ℓ − ℓ 0 ) 2 ≤ 1/4
1 + λ ∗ ℓ r ≤ 1 + r 2 2 T + r 2 T ≤ C ( 1 + r 2 / T ) . 1+\lambda_*\ell r\le1+\frac{r^2}{2T}+\frac{r}{2\sqrt{T}}\le C(1+r^2/T). 1 + λ ∗ ℓ r ≤ 1 + 2 T r 2 + 2 T r ≤ C ( 1 + r 2 / T ) . Meanwhile,
e λ ∗ z σ ( − T , x ) = e − 2 f ′ ( 0 ) T L σ ( r e 1 ) ≤ C ( 1 + r ) − N − 1 2 e − 2 f ′ ( 0 ) T + λ ∗ r . e^{\lambda_*z_\sigma(-T,x)}=e^{-2f'(0)TL_\sigma(re_1)}\le C(1+r)^{-\frac{N-1}{2}}e^{-2f'(0)T+\lambda_*r}. e λ ∗ z σ ( − T , x ) = e − 2 f ′ ( 0 ) T L σ ( r e 1 ) ≤ C ( 1 + r ) − 2 N − 1 e − 2 f ′ ( 0 ) T + λ ∗ r . Since R ( − T ) ≍ T R(-T)\asymp T R ( − T ) ≍ T T → ∞ T\to\infty T → ∞ 2 (101) ,
H ( − T , x ) T e λ ∗ z σ ( − T , x ) ≥ c T − p − 3 / 2 ( 1 + r 1 + r 2 / T ) N − 1 2 exp ( ( c ∗ T − r ) 2 4 T ) , (102) \frac{H(-T,x)}{Te^{\lambda_*z_\sigma(-T,x)}}\ge cT^{-p-3/2}\left(\frac{1+r}{1+r^2/T}\right)^{\frac{N-1}{2}}\exp\left(\frac{(c_*T-r)^2}{4T}\right), \tag*{(102)} T e λ ∗ z σ ( − T , x ) H ( − T , x ) ≥ c T − p − 3/2 ( 1 + r 2 / T 1 + r ) 2 N − 1 exp ( 4 T ( c ∗ T − r ) 2 ) , (102) β ( − T ) H ( − T , x ) T e λ ∗ z σ ( − T , x ) ≥ c T − p − 5 / 2 ( 1 + r 1 + r 2 / T ) N − 1 2 exp ( ( c ∗ T − r ) 2 4 T ) . (103) \frac{\beta(-T)H(-T,x)}{Te^{\lambda_*z_\sigma(-T,x)}}\ge cT^{-p-5/2}\left(\frac{1+r}{1+r^2/T}\right)^{\frac{N-1}{2}}\exp\left(\frac{(c_*T-r)^2}{4T}\right). \tag*{(103)} T e λ ∗ z σ ( − T , x ) β ( − T ) H ( − T , x ) ≥ c T − p − 5/2 ( 1 + r 2 / T 1 + r ) 2 N − 1 exp ( 4 T ( c ∗ T − r ) 2 ) . (103) Here r ≤ c ∗ T r\le c_*T r ≤ c ∗ T ( 1 + r ) / ( 1 + r 2 / T ) ≥ min { 1 , c ∗ − 1 } > 0 (1+r)/(1+r^2/T)\ge\min\{1,c_*^{-1}\}>0 ( 1 + r ) / ( 1 + r 2 / T ) ≥ min { 1 , c ∗ − 1 } > 0 c ∗ T − r ≥ c 0 T / ( ln T ) 2 c_*T-r\ge c_0T/(\ln T)^2 c ∗ T − r ≥ c 0 T / ( ln T ) 2 (101) , we also have
( c ∗ T − r ) 2 4 T ≥ c T ( ln T ) 4 . \frac{(c_*T-r)^2}{4T}\ge\frac{cT}{(\ln T)^4}. 4 T ( c ∗ T − r ) 2 ≥ ( ln T ) 4 c T . Both right sides in (102) –(103) consequently tend to infinity as T → ∞ T\to\infty T → ∞ d ( − T , x ) ≤ − L ( − T ) d(-T,x)\le-L(-T) d ( − T , x ) ≤ − L ( − T )
On d ( − T , x ) ≤ − L ( − T ) d(-T,x)\le-L(-T) d ( − T , x ) ≤ − L ( − T ) − c ∗ T ≤ z σ ( − T , x ) ≤ − 1 -c_*T\le z_\sigma(-T,x)\le-1 − c ∗ T ≤ z σ ( − T , x ) ≤ − 1 T T T (6) gives u ≤ C T e λ ∗ z σ u\le CTe^{\lambda_*z_\sigma} u ≤ C T e λ ∗ z σ (102) proves
H ≥ u for d ≤ − L ( t ) . (104) H\ge u\qquad\text{for }d\le-L(t). \tag*{(104)} H ≥ u for d ≤ − L ( t ) . (104) For − 2 L ( t ) ≤ d ≤ 0 -2L(t)\le d\le0 − 2 L ( t ) ≤ d ≤ 0
z σ ( t , x ) − d ( t , x ) = ∫ R ( t ) + d ( t , x ) R ( t ) ( 1 − ∂ r z σ ( t , s ) ) d s z_\sigma(t,x)-d(t,x)=\int_{R(t)+d(t,x)}^{R(t)}(1-\partial_r z_\sigma(t,s))\,ds z σ ( t , x ) − d ( t , x ) = ∫ R ( t ) + d ( t , x ) R ( t ) ( 1 − ∂ r z σ ( t , s )) d s and (93) give
0 ≤ z σ − d ≤ C ∣ d ∣ / R ( t ) ≤ C / ( ln R ( t ) ) 2 . (105) 0\le z_\sigma-d\le C|d|/R(t)\le C/(\ln R(t))^2. \tag*{(105)} 0 ≤ z σ − d ≤ C ∣ d ∣/ R ( t ) ≤ C / ( ln R ( t ) ) 2 . (105) Since e λ ∗ d ≤ e λ ∗ z σ e^{\lambda_*d}\le e^{\lambda_*z_\sigma} e λ ∗ d ≤ e λ ∗ z σ 1 + L ( t ) ≤ C ∣ t ∣ 1+L(t)\le C|t| 1 + L ( t ) ≤ C ∣ t ∣ (103) and (98) give, for T 0 T_0 T 0
F [ Ψ ] + β H ≥ 0 for − 2 L ( t ) ≤ d ≤ − L ( t ) , t ≤ − T 0 . (106) \mathcal{F}[\Psi]+\beta H\ge0\qquad\text{for }-2L(t)\le d\le-L(t),\ t\le-T_0. \tag*{(106)} F [ Ψ ] + β H ≥ 0 for − 2 L ( t ) ≤ d ≤ − L ( t ) , t ≤ − T 0 . (106) Define, for t ≤ − T 0 t\le-T_0 t ≤ − T 0
u ‾ ( t , x ) : = min { 1 , Ψ ( t , x ) + H ( t , x ) } . \overline{u}(t,x):=\min\{1,\Psi(t,x)+H(t,x)\}. u ( t , x ) := min { 1 , Ψ ( t , x ) + H ( t , x )} . This is a bounded supersolution. Indeed, writing W = Ψ + H W=\Psi+H W = Ψ + H { W < 1 } \{W<1\} { W < 1 }
F [ W ] = F [ Ψ ] + β H + f ′ ( 0 ) H + f ( Ψ ) − f ( Ψ + H ) ≥ F [ Ψ ] + β H ≥ 0. \mathcal{F}[W]=\mathcal{F}[\Psi]+\beta H+f'(0)H+f(\Psi)-f(\Psi+H)
\ge\mathcal{F}[\Psi]+\beta H\ge0. F [ W ] = F [ Ψ ] + β H + f ′ ( 0 ) H + f ( Ψ ) − f ( Ψ + H ) ≥ F [ Ψ ] + β H ≥ 0. The first inequality follows by integrating f ′ ≤ f ′ ( 0 ) f' \le f'(0) f ′ ≤ f ′ ( 0 ) Ψ \Psi Ψ Ψ + H \Psi+H Ψ + H (106) on the cutoff strip, from (97) where d ≥ − L d \ge-L d ≥ − L Ψ = 0 \Psi=0 Ψ = 0 d ≤ − 2 L d \le-2L d ≤ − 2 L
Here is a distributional verification of the cap. Put L = ∂ t − Δ \mathcal{L}=\partial_t-\Delta L = ∂ t − Δ v v v v + = max { v , 0 } v_+=\max\{v,0\} v + = max { v , 0 }
L ( v + ) ≤ 1 { v > 0 } L v \mathcal{L}(v_+) \le\mathbf{1}_{\{v>0\}}\mathcal{L}v L ( v + ) ≤ 1 { v > 0 } L v in distributions. More explicitly, choose increasing convex θ ε \theta_\varepsilon θ ε s ≤ 0 s \le0 s ≤ 0 s + s_+ s +
L θ ε ( v ) = θ ε ′ ( v ) L v − θ ε ′ ′ ( v ) ∣ ∇ v ∣ 2 ≤ θ ε ′ ( v ) L v , \mathcal{L}\theta_\varepsilon(v)=\theta_\varepsilon'(v)\mathcal{L}v-\theta_\varepsilon''(v)|\nabla v|^2\le\theta_\varepsilon'(v)\mathcal{L}v, L θ ε ( v ) = θ ε ′ ( v ) L v − θ ε ′′ ( v ) ∣∇ v ∣ 2 ≤ θ ε ′ ( v ) L v , and pass to the distributional limit. Applied to v = 1 − W v=1-W v = 1 − W
L u ‾ = − L ( 1 − W ) + ≥ 1 { W < 1 } L W ≥ 1 { W < 1 } f ( W ) = f ( u ‾ ) . \mathcal{L}\overline{u}=-\mathcal{L}(1-W)_+\ge\mathbf{1}_{\{W<1\}}\mathcal{L}W\ge\mathbf{1}_{\{W<1\}}f(W)=f(\overline{u}). L u = − L ( 1 − W ) + ≥ 1 { W < 1 } L W ≥ 1 { W < 1 } f ( W ) = f ( u ) . The last equality uses f ( 1 ) = 0 f(1)=0 f ( 1 ) = 0 H H H 0 ≤ u ‾ ≤ 1 0\le\overline{u}\le1 0 ≤ u ≤ 1
We verify u ≤ u ‾ u\le\overline{u} u ≤ u d ≤ − L ( t ) d\le-L(t) d ≤ − L ( t ) (104) . On − L ( t ) ≤ d ≤ 0 -L(t)\le d\le0 − L ( t ) ≤ d ≤ 0 (105) and η ( t ) = D / ln R ( t ) \eta(t)=D/\ln R(t) η ( t ) = D / ln R ( t ) z σ ≤ d + η ( t ) z_\sigma\le d+\eta(t) z σ ≤ d + η ( t ) R ( t ) R(t) R ( t ) Ψ = v \Psi=v Ψ = v d ≥ 0 d\ge0 d ≥ 0 ∂ r z σ ( t , r ) ≤ 1 \partial_r z_\sigma(t,r)\le1 ∂ r z σ ( t , r ) ≤ 1 z σ ≤ d z_\sigma\le d z σ ≤ d u ≤ u ‾ u\le\overline{u} u ≤ u d ≥ − L ( t ) d\ge-L(t) d ≥ − L ( t )
0 < u ≤ u ‾ ≤ 1 for t ≤ − T 0 . (107) 0<u\le\overline{u}\le1\qquad\text{for }t\le-T_0. \tag*{(107)} 0 < u ≤ u ≤ 1 for t ≤ − T 0 . (107) Both u u u u ‾ \overline{u} u x x x n > T 0 n>T_0 n > T 0 u n u_n u n u n ( − n , x ) = u ( − n , x ) u_n(-n,x)=u(-n,x) u n ( − n , x ) = u ( − n , x ) u n u_n u n x x x
u ≤ u n ≤ u ‾ ≤ 1 for − n ≤ t ≤ − T 0 , 0 ≤ u n ≤ 1 for t ≥ − n . u\le u_n\le\overline{u}\le1\qquad\text{for }-n\le t\le-T_0,\qquad0\le u_n\le1\qquad\text{for }t\ge-n. u ≤ u n ≤ u ≤ 1 for − n ≤ t ≤ − T 0 , 0 ≤ u n ≤ 1 for t ≥ − n . At time − n -n − n u n + 1 ( − n , ⋅ ) ≥ u ( − n , ⋅ ) = u n ( − n , ⋅ ) u_{n+1}(-n,\cdot)\ge u(-n,\cdot)=u_n(-n,\cdot) u n + 1 ( − n , ⋅ ) ≥ u ( − n , ⋅ ) = u n ( − n , ⋅ ) u n + 1 ( t , x ) ≥ u n ( t , x ) u_{n+1}(t,x)\ge u_n(t,x) u n + 1 ( t , x ) ≥ u n ( t , x ) t ≥ − n t\ge-n t ≥ − n x ∈ R N x\in\mathbb{R}^N x ∈ R N
u n ↗ U r a d in C l o c 1 , 2 ( R × R N ) as n → ∞ . u_n \nearrow U^{\mathrm{rad}}\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N)\quad\text{as }n\to\infty. u n ↗ U rad in C loc 1 , 2 ( R × R N ) as n → ∞. The limit U r a d U^{\mathrm{rad}} U rad u ≤ U r a d ≤ u ‾ u\le U^{\mathrm{rad}}\le\overline{u} u ≤ U rad ≤ u t ≤ − T 0 t\le-T_0 t ≤ − T 0
For each fixed M M M Ψ ( − T , x ) \Psi(-T,x) Ψ ( − T , x ) ∣ x ∣ ≤ M |x|\le M ∣ x ∣ ≤ M T T T R ( − T ) − 2 L ( − T ) → ∞ R(-T)-2L(-T)\to\infty R ( − T ) − 2 L ( − T ) → ∞ (100) gives
sup ∣ x ∣ ≤ M U r a d ( − T , x ) ≤ C M T − p − 1 / 2 e − f ′ ( 0 ) T → 0. (108) \sup_{|x|\le M}U^{\mathrm{rad}}(-T,x)\le C_M T^{-p-1/2}e^{-f'(0)T}\to0. \tag*{(108)} ∣ x ∣ ≤ M sup U rad ( − T , x ) ≤ C M T − p − 1/2 e − f ′ ( 0 ) T → 0. (108) The lower bound by u u u 1 1 1 U r a d U^{\mathrm{rad}} U rad 1 1 1 1 − U r a d 1-U^{\mathrm{rad}} 1 − U rad 1 1 1 (108) . Indeed, w = 1 − U r a d w=1-U^{\mathrm{rad}} w = 1 − U rad w t − Δ w = c ( t , x ) w w_t-\Delta w=c(t,x)w w t − Δ w = c ( t , x ) w c = − f ( U r a d ) / ( 1 − U r a d ) c=-f(U^{\mathrm{rad}})/(1-U^{\mathrm{rad}}) c = − f ( U rad ) / ( 1 − U rad ) U r a d = 1 U^{\mathrm{rad}}=1 U rad = 1 f ′ ( 1 ) f'(1) f ′ ( 1 )
0 < U r a d < 1 , U r a d ( t , 0 ) → 0 ( t → − ∞ ) . (109) 0<U^{\mathrm{rad}}<1,\qquad U^{\mathrm{rad}}(t,0)\to0\qquad(t\to-\infty). \tag*{(109)} 0 < U rad < 1 , U rad ( t , 0 ) → 0 ( t → − ∞ ) . (109) By the comparison principle, U r a d U^{\mathrm{rad}} U rad u = ϕ ∗ ( z σ ) u=\phi_*(z_\sigma) u = ϕ ∗ ( z σ ) σ \sigma σ u σ : = U r a d u_\sigma:=U^{\mathrm{rad}} u σ := U rad
Finally, U r a d U^{\mathrm{rad}} U rad h > 0 h>0 h > 0
u n ( − n + h , ⋅ ) ≥ u ( − n + h , ⋅ ) ≥ u ( − n , ⋅ ) . u_n(-n+h,\cdot)\ge u(-n+h,\cdot)\ge u(-n,\cdot). u n ( − n + h , ⋅ ) ≥ u ( − n + h , ⋅ ) ≥ u ( − n , ⋅ ) . Comparing u n ( t + h , ⋅ ) u_n(t+h,\cdot) u n ( t + h , ⋅ ) u n ( t , ⋅ ) u_n(t,\cdot) u n ( t , ⋅ ) U r a d ( t + h , x ) ≥ U r a d ( t , x ) U^{\mathrm{rad}}(t+h,x)\ge U^{\mathrm{rad}}(t,x) U rad ( t + h , x ) ≥ U rad ( t , x ) v = ∂ t U r a d ≥ 0 v=\partial_tU^{\mathrm{rad}}\ge0 v = ∂ t U rad ≥ 0
v t − Δ v = f ′ ( U r a d ) v . v_t-\Delta v=f'(U^{\mathrm{rad}})v. v t − Δ v = f ′ ( U rad ) v . If v v v ∂ t U r a d > 0 \partial_tU^{\mathrm{rad}}>0 ∂ t U rad > 0
For the measure b σ b\sigma bσ b > 0 b>0 b > 0 u σ u_\sigma u σ
ϕ ∗ ( z b σ ( t , x ) ) = ϕ ∗ ( z σ ( t , x ) + λ ∗ − 1 ln b ) = u ( t + ln b 2 f ′ ( 0 ) , x ) . \phi_*(z_{b\sigma}(t,x))=\phi_*(z_\sigma(t,x)+\lambda_*^{-1}\ln b)=u\left(t+\frac{\ln b}{2f'(0)},x\right). ϕ ∗ ( z bσ ( t , x )) = ϕ ∗ ( z σ ( t , x ) + λ ∗ − 1 ln b ) = u ( t + 2 f ′ ( 0 ) ln b , x ) . Thus U r a d ( t + ln b / ( 2 f ′ ( 0 ) ) , x ) U^{\mathrm{rad}}(t+\ln b/(2f'(0)),x) U rad ( t + ln b / ( 2 f ′ ( 0 )) , x ) ϕ ∗ ( z b σ ( t , x ) ) \phi_*(z_{b\sigma}(t,x)) ϕ ∗ ( z bσ ( t , x )) W ∈ E W\in\mathcal{E} W ∈ E W ≥ ϕ ∗ ( z b σ ) W\ge\phi_*(z_{b\sigma}) W ≥ ϕ ∗ ( z bσ )
W ( t − ln b 2 f ′ ( 0 ) , x ) ≥ u ( t , x ) . W\left(t-\frac{\ln b}{2f'(0)},x\right)\ge u(t,x). W ( t − 2 f ′ ( 0 ) ln b , x ) ≥ u ( t , x ) . Minimality of U r a d U^{\mathrm{rad}} U rad
U r a d ( t + ln b 2 f ′ ( 0 ) , x ) ≤ W ( t , x ) . U^{\mathrm{rad}}\left(t+\frac{\ln b}{2f'(0)},x\right)\le W(t,x). U rad ( t + 2 f ′ ( 0 ) ln b , x ) ≤ W ( t , x ) . Hence this time translate is the minimal entire solution above ϕ ∗ ( z b σ ) \phi_*(z_{b\sigma}) ϕ ∗ ( z bσ )
u b σ ( t , x ) : = u σ ( t + ln b 2 f ′ ( 0 ) , x ) = U r a d ( t + ln b 2 f ′ ( 0 ) , x ) . (110) u_{b\sigma}(t,x):=u_\sigma\left(t+\frac{\ln b}{2f'(0)},x\right)=U^{\mathrm{rad}}\left(t+\frac{\ln b}{2f'(0)},x\right). \tag*{(110)} u bσ ( t , x ) := u σ ( t + 2 f ′ ( 0 ) ln b , x ) = U rad ( t + 2 f ′ ( 0 ) ln b , x ) . (110) This completes the proof.
Entire solutions from critical-sphere measures
We now construct a canonical entire solution for each finite positive measure on the critical sphere. Spherical averaging provides a radial upper bound, and comparison relates scaling of the measure to dilation and time translation of the solution.
Let g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) g(q)=\phi_*(\lambda_*^{-1}\ln q) g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) q > 0 q>0 q > 0 g ( 0 ) = 0 g(0)=0 g ( 0 ) = 0 Lemma 3.6 . Let σ \sigma σ S N − 1 S^{N-1} S N − 1 N = 1 N=1 N = 1 ( δ + 1 + δ − 1 ) / 2 (\delta_{+1}+\delta_{-1})/2 ( δ + 1 + δ − 1 ) /2 μ \mu μ L μ L_\mu L μ z μ z_\mu z μ (12) and set
H μ ( t , x ) : = e 2 f ′ ( 0 ) t L μ ( x ) = e λ ∗ z μ ( t , x ) . (111) H_\mu(t,x):=e^{2f'(0)t}L_\mu(x)=e^{\lambda_*z_\mu(t,x)}. \tag*{(111)} H μ ( t , x ) := e 2 f ′ ( 0 ) t L μ ( x ) = e λ ∗ z μ ( t , x ) . (111) We use the radial solution to ensure that the construction for an arbitrary finite positive measure stays strictly below one. We also identify the action of dilation on the measure.
Theorem 5.2. For every finite positive Borel measure μ \mu μ S N − 1 S^{N-1} S N − 1 u μ u_\mu u μ (1) satisfying
0 < ϕ ∗ ( z μ ( t , x ) ) ≤ u μ ( t , x ) ≤ U r a d ( z μ ( t , x ) c ∗ , 0 ) < 1 (112) 0<\phi_*(z_\mu(t,x))\le u_\mu(t,x)\le U^{\mathrm{rad}}\left(\frac{z_\mu(t,x)}{c_*},0\right)<1 \tag*{(112)} 0 < ϕ ∗ ( z μ ( t , x )) ≤ u μ ( t , x ) ≤ U rad ( c ∗ z μ ( t , x ) , 0 ) < 1 (112) for every ( t , x ) ∈ R × R N (t,x)\in\mathbb{R}\times\mathbb{R}^N ( t , x ) ∈ R × R N
u μ ( t , ⋅ ) → 0 locally uniformly in R N as t → − ∞ . u_\mu(t,\cdot)\to0\qquad\text{locally uniformly in }\mathbb{R}^N\quad\text{as }t\to-\infty. u μ ( t , ⋅ ) → 0 locally uniformly in R N as t → − ∞. Moreover,
( D b u μ ) ( t , x ) = u b μ ( t , x ) = u μ ( t + ln b 2 f ′ ( 0 ) , x ) for b > 0. (113) (D_bu_\mu)(t,x)=u_{b\mu}(t,x)=u_\mu\left(t+\frac{\ln b}{2f'(0)},x\right)\qquad\text{for }b>0. \tag*{(113)} ( D b u μ ) ( t , x ) = u b μ ( t , x ) = u μ ( t + 2 f ′ ( 0 ) ln b , x ) for b > 0. (113) Proof. These functions and all their derivatives are finite on compact space–time sets. They satisfy H t − Δ H = f ′ ( 0 ) H H_t-\Delta H=f'(0)H H t − Δ H = f ′ ( 0 ) H ∣ ∇ H ∣ ≤ λ ∗ H |\nabla H|\leq\lambda_*H ∣∇ H ∣ ≤ λ ∗ H
( ∂ t − Δ ) g ( H ) − f ( g ( H ) ) = g ′ ′ ( H ) ( f ′ ( 0 ) H 2 − ∣ ∇ H ∣ 2 ) ≤ 0. (\partial_t-\Delta)g(H)-f(g(H))=g''(H)\left(f'(0)H^2-|\nabla H|^2\right)\leq0. ( ∂ t − Δ ) g ( H ) − f ( g ( H )) = g ′′ ( H ) ( f ′ ( 0 ) H 2 − ∣∇ H ∣ 2 ) ≤ 0. Thus g ( H μ ) g(H_\mu) g ( H μ ) ( t , x ) (t,x) ( t , x ) ( S t + T [ g ( H μ ( − T , ⋅ ) ) ] ) ( x ) (S_{t+T}[g(H_\mu(-T,\cdot))])(x) ( S t + T [ g ( H μ ( − T , ⋅ ))]) ( x ) T > max { 0 , − t } T>\max\{0,-t\} T > max { 0 , − t }
u μ ( t , x ) : = lim T → ∞ ( S t + T [ g ( H μ ( − T , ⋅ ) ) ] ) ( x ) (114) u_\mu(t,x):=\lim_{T\to\infty}(S_{t+T}[g(H_\mu(-T,\cdot))])(x) \tag*{(114)} u μ ( t , x ) := T → ∞ lim ( S t + T [ g ( H μ ( − T , ⋅ ))]) ( x ) (114) defines an entire solution of (1) . The convergence holds in C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) u 0 : = 0 u_0:=0 u 0 := 0 Z Z Z g ( H μ ) g(H_\mu) g ( H μ ) − T -T − T Z ≥ u μ Z\geq u_\mu Z ≥ u μ
For N = 1 N=1 N = 1 μ = α δ + 1 + β δ − 1 \mu=\alpha\delta_{+1}+\beta\delta_{-1} μ = α δ + 1 + β δ − 1 α , β ≥ 0 \alpha,\beta\geq0 α , β ≥ 0 α + β > 0 \alpha+\beta>0 α + β > 0
v + ( t , x ) = ϕ ∗ ( x + c ∗ t + λ ∗ − 1 ln α ) , v − ( t , x ) = ϕ ∗ ( − x + c ∗ t + λ ∗ − 1 ln β ) , v_+(t,x)=\phi_*(x+c_*t+\lambda_*^{-1}\ln\alpha),\qquad v_-(t,x)=\phi_*(-x+c_*t+\lambda_*^{-1}\ln\beta), v + ( t , x ) = ϕ ∗ ( x + c ∗ t + λ ∗ − 1 ln α ) , v − ( t , x ) = ϕ ∗ ( − x + c ∗ t + λ ∗ − 1 ln β ) , for α > 0 \alpha>0 α > 0 β > 0 \beta>0 β > 0 v + ≡ 0 v_+\equiv0 v + ≡ 0 α = 0 \alpha=0 α = 0 v − ≡ 0 v_-\equiv0 v − ≡ 0 β = 0 \beta=0 β = 0 g g g g ( 0 ) = 0 g(0)=0 g ( 0 ) = 0 max { v + , v − } ≤ g ( H μ ) ≤ min { v + + v − , 1 } \max\{v_+,v_-\}\leq g(H_\mu)\leq\min\{v_++v_-,1\} max { v + , v − } ≤ g ( H μ ) ≤ min { v + + v − , 1 } (114) therefore yields
max { v + , v − } ≤ u μ ≤ min { v + + v − , 1 } . \max\{v_+,v_-\}\leq u_\mu\leq\min\{v_++v_-,1\}. max { v + , v − } ≤ u μ ≤ min { v + + v − , 1 } . By Theorem 3.4 , u μ u_\mu u μ [28 Theorem 1.1] . In particular, 0 < u μ < 1 0<u_\mu<1 0 < u μ < 1 μ = σ = ( δ + 1 + δ − 1 ) / 2 \mu=\sigma=(\delta_{+1}+\delta_{-1})/2 μ = σ = ( δ + 1 + δ − 1 ) /2 U r a d = u σ U^{\mathrm{rad}}=u_\sigma U rad = u σ ∂ t U r a d > 0 \partial_tU^{\mathrm{rad}}>0 ∂ t U rad > 0 (110) in this dimension.
We next use the radially symmetric solution U r a d U^{\mathrm{rad}} U rad u μ ( t , x ) < 1 u_\mu(t,x)<1 u μ ( t , x ) < 1 μ \mu μ ( t , x ) ∈ R × R N (t,x)\in\mathbb{R}\times\mathbb{R}^N ( t , x ) ∈ R × R N b > 0 b>0 b > 0 (111) gives
H b μ ( t , x ) = b H μ ( t , x ) = H μ ( t + ln b 2 f ′ ( 0 ) , x ) . H_{b\mu}(t,x)=bH_\mu(t,x)=H_\mu\left(t+\frac{\ln b}{2f'(0)},x\right). H b μ ( t , x ) = b H μ ( t , x ) = H μ ( t + 2 f ′ ( 0 ) ln b , x ) . Substituting this identity into (114) , we obtain
u b μ ( t , x ) = lim T → ∞ ( S t + T [ g ( H μ ( − T + ln b 2 f ′ ( 0 ) , ⋅ ) ) ] ) ( x ) = u μ ( t + ln b 2 f ′ ( 0 ) , x ) . (115) \begin{aligned}
u_{b\mu}(t,x)&=\lim_{T\to\infty}\left(S_{t+T}\left[g\left(H_\mu\left(-T+\frac{\ln b}{2f'(0)},\cdot\right)\right)\right]\right)(x)\\
&=u_\mu\left(t+\frac{\ln b}{2f'(0)},x\right). \tag*{(115)}
\end{aligned} u b μ ( t , x ) = T → ∞ lim ( S t + T [ g ( H μ ( − T + 2 f ′ ( 0 ) ln b , ⋅ ) ) ] ) ( x ) = u μ ( t + 2 f ′ ( 0 ) ln b , x ) . (115) Let M = μ ( S N − 1 ) M=\mu(\mathbb{S}^{N-1}) M = μ ( S N − 1 ) T T T L σ L_\sigma L σ
∫ S N − 1 L μ ( ∣ x ∣ ξ ) d σ ( ξ ) = ∫ S N − 1 ( ∫ S N − 1 e λ ∗ ∣ x ∣ ξ ⋅ ν d σ ( ξ ) ) d μ ( ν ) = ∫ S N − 1 L σ ( ∣ x ∣ ν ) d μ ( ν ) = M L σ ( x ) . (116) \begin{aligned}
\int_{\mathbb{S}^{N-1}}L_\mu(|x|\xi)\,d\sigma(\xi)
&=\int_{\mathbb{S}^{N-1}}\left(\int_{\mathbb{S}^{N-1}}e^{\lambda_*|x|\xi\cdot\nu}\,d\sigma(\xi)\right)d\mu(\nu)\\
&=\int_{\mathbb{S}^{N-1}}L_\sigma(|x|\nu)\,d\mu(\nu)=ML_\sigma(x). \tag*{(116)}
\end{aligned} ∫ S N − 1 L μ ( ∣ x ∣ ξ ) d σ ( ξ ) = ∫ S N − 1 ( ∫ S N − 1 e λ ∗ ∣ x ∣ ξ ⋅ ν d σ ( ξ ) ) d μ ( ν ) = ∫ S N − 1 L σ ( ∣ x ∣ ν ) d μ ( ν ) = M L σ ( x ) . (116) Concavity of g g g
∫ S N − 1 g ( H μ ( − T , ∣ x ∣ ξ ) ) d σ ( ξ ) ≤ g ( H M σ ( − T , x ) ) . \int_{\mathbb{S}^{N-1}}g(H_\mu(-T,|x|\xi))\,d\sigma(\xi)\leq g(H_{M\sigma}(-T,x)). ∫ S N − 1 g ( H μ ( − T , ∣ x ∣ ξ )) d σ ( ξ ) ≤ g ( H M σ ( − T , x )) . Set
v μ , T ( t , x ) : = ( S t + T [ g ( H μ ( − T , ⋅ ) ) ] ) ( x ) for t ≥ − T , v_{\mu,T}(t,x):=(S_{t+T}[g(H_\mu(-T,\cdot))])(x)\qquad\text{for }t\geq-T, v μ , T ( t , x ) := ( S t + T [ g ( H μ ( − T , ⋅ ))]) ( x ) for t ≥ − T , and define its spherical average by
W T ( t , x ) : = ∫ S N − 1 v μ , T ( t , ∣ x ∣ ξ ) d σ ( ξ ) . W_T(t,x) := \int_{\mathbb{S}^{N-1}} v_{\mu,T}(t,|x|\xi)\,d\sigma(\xi). W T ( t , x ) := ∫ S N − 1 v μ , T ( t , ∣ x ∣ ξ ) d σ ( ξ ) . Since spherical averaging commutes with the Laplacian, concavity of f f f
( W T ) t − Δ W T = ∫ S N − 1 f ( v μ , T ( t , ∣ x ∣ ξ ) ) d σ ( ξ ) ≤ f ( W T ) . (W_T)_t-\Delta W_T=\int_{\mathbb{S}^{N-1}}f(v_{\mu,T}(t,|x|\xi))\,d\sigma(\xi)\leq f(W_T). ( W T ) t − Δ W T = ∫ S N − 1 f ( v μ , T ( t , ∣ x ∣ ξ )) d σ ( ξ ) ≤ f ( W T ) . Thus it is a subsolution. Comparison gives W T ≤ v M σ , T W_T\leq v_{M\sigma,T} W T ≤ v M σ , T W T ( t , 0 ) = v μ , T ( t , 0 ) W_T(t,0)=v_{\mu,T}(t,0) W T ( t , 0 ) = v μ , T ( t , 0 ) T → ∞ T\to\infty T → ∞
u μ ( t , 0 ) ≤ u M σ ( t , 0 ) for t ∈ R . (117) u_\mu(t,0)\leq u_{M\sigma}(t,0)\qquad\text{for }t\in\mathbb{R}. \tag*{(117)} u μ ( t , 0 ) ≤ u M σ ( t , 0 ) for t ∈ R . (117) For a general point x x x d μ x ( ν ) = e λ ∗ ⋅ x ⋅ ν d μ ( ν ) d\mu_x(\nu)=e^{\lambda_*\cdot x\cdot\nu}d\mu(\nu) d μ x ( ν ) = e λ ∗ ⋅ x ⋅ ν d μ ( ν ) L μ x ( y ) = L μ ( x + y ) L_{\mu_x}(y)=L_\mu(x+y) L μ x ( y ) = L μ ( x + y ) L μ ( x ) L_\mu(x) L μ ( x ) u μ x ( t , y ) = u μ ( t , x + y ) u_{\mu_x}(t,y)=u_\mu(t,x+y) u μ x ( t , y ) = u μ ( t , x + y ) (117) to μ x \mu_x μ x L μ ( x ) L_\mu(x) L μ ( x )
u μ ( t , x ) = u μ x ( t , 0 ) ≤ u L μ ( x ) σ ( t , 0 ) . u_\mu(t,x)=u_{\mu_x}(t,0)\leq u_{L_\mu(x)\sigma}(t,0). u μ ( t , x ) = u μ x ( t , 0 ) ≤ u L μ ( x ) σ ( t , 0 ) . By (115) with b = L μ ( x ) b=L_\mu(x) b = L μ ( x ) u σ = U r a d u_\sigma=U^{\mathrm{rad}} u σ = U rad
u L μ ( x ) σ ( t , 0 ) = U r a d ( t + ln L μ ( x ) 2 f ′ ( 0 ) , 0 ) = U r a d ( z μ ( t , x ) c ∗ , 0 ) . u_{L_\mu(x)\sigma}(t,0)=U^{\mathrm{rad}}\left(t+\frac{\ln L_\mu(x)}{2f'(0)},0\right)=U^{\mathrm{rad}}\left(\frac{z_\mu(t,x)}{c_*},0\right). u L μ ( x ) σ ( t , 0 ) = U rad ( t + 2 f ′ ( 0 ) ln L μ ( x ) , 0 ) = U rad ( c ∗ z μ ( t , x ) , 0 ) . Together with the lower bound from (114) and U r a d < 1 U^{\mathrm{rad}}<1 U rad < 1 (112) . The same bound and the past decay of U r a d U^{\mathrm{rad}} U rad u μ ( t , ⋅ ) → 0 u_\mu(t,\cdot)\to0 u μ ( t , ⋅ ) → 0 R N \mathbb{R}^N R N t → − ∞ t\to-\infty t → − ∞
We next prove that D b u μ = u b μ D_bu_\mu=u_{b\mu} D b u μ = u b μ b > 0 b>0 b > 0 u μ u_\mu u μ b b b (6) , and Lemma 3.6 give
d 2 d s 2 ( e − λ ∗ s ϕ ∗ ( s ) ) = e − λ ∗ s ( f ′ ( 0 ) ϕ ∗ ( s ) − f ( ϕ ∗ ( s ) ) ) ≥ 0 , − 1 ≤ d d s ( e − λ ∗ s ϕ ∗ ( s ) ) ≤ 0. \frac{d^2}{ds^2}\left(e^{-\lambda_*s}\phi_*(s)\right)=e^{-\lambda_*s}\left(f'(0)\phi_*(s)-f(\phi_*(s))\right)\geq0,\qquad-1\leq\frac{d}{ds}\left(e^{-\lambda_*s}\phi_*(s)\right)\leq0. d s 2 d 2 ( e − λ ∗ s ϕ ∗ ( s ) ) = e − λ ∗ s ( f ′ ( 0 ) ϕ ∗ ( s ) − f ( ϕ ∗ ( s )) ) ≥ 0 , − 1 ≤ d s d ( e − λ ∗ s ϕ ∗ ( s ) ) ≤ 0. For b > 0 b>0 b > 0 q > 0 q>0 q > 0
∣ b g ( q ) − g ( b q ) ∣ = b q ∣ ∫ λ ∗ − 1 ln q λ ∗ − 1 ln ( b q ) d d r ( e − λ ∗ r ϕ ∗ ( r ) ) d r ∣ ≤ b λ ∗ q ∣ ln b ∣ . (118) \left|bg(q)-g(bq)\right|=bq\left|\int_{\lambda_*^{-1}\ln q}^{\lambda_*^{-1}\ln(bq)}\frac{d}{dr}\left(e^{-\lambda_*r}\phi_*(r)\right)\,dr\right|\leq\frac{b}{\lambda_*}q|\ln b|. \tag*{(118)} ∣ b g ( q ) − g ( b q ) ∣ = b q ∫ λ ∗ − 1 l n q λ ∗ − 1 l n ( b q ) d r d ( e − λ ∗ r ϕ ∗ ( r ) ) d r ≤ λ ∗ b q ∣ ln b ∣. (118) The left side is also bounded by b + 1 b+1 b + 1 min { b λ ∗ ∣ ln b ∣ H μ , b + 1 } \min\left\{\frac{b}{\lambda_*}|\ln b|H_\mu,b+1\right\} min { λ ∗ b ∣ ln b ∣ H μ , b + 1 } q = H μ q=H_\mu q = H μ
Fix b > 0 b>0 b > 0 b ≠ 1 b\ne1 b = 1 (39) and (118) , and using S t [ B h ] ≤ B S t [ h ] S_t[Bh]\leq BS_t[h] S t [ B h ] ≤ B S t [ h ] B ≥ 1 B\geq1 B ≥ 1
∣ S t + T [ b g ( H μ ( − T , ⋅ ) ) ] − S t + T [ g ( b H μ ( − T , ⋅ ) ) ] ∣ ≤ S t + T [ min { b λ ∗ ∣ ln b ∣ H μ ( − T , ⋅ ) , b + 1 } ] ≤ ( b + 1 ) S t + T [ min { b ∣ ln b ∣ λ ∗ ( b + 1 ) H μ ( − T , ⋅ ) , 1 } ] → 0 \begin{aligned}
\left|S_{t+T}[bg(H_\mu(-T,\cdot))]-S_{t+T}[g(bH_\mu(-T,\cdot))]\right|
&\leq S_{t+T}\left[\min\left\{\frac{b}{\lambda_*}|\ln b|H_\mu(-T,\cdot),b+1\right\}\right]\\
&\leq(b+1)S_{t+T}\left[\min\left\{\frac{b|\ln b|}{\lambda_*(b+1)}H_\mu(-T,\cdot),1\right\}\right]\to0
\end{aligned} ∣ S t + T [ b g ( H μ ( − T , ⋅ ))] − S t + T [ g ( b H μ ( − T , ⋅ ))] ∣ ≤ S t + T [ min { λ ∗ b ∣ ln b ∣ H μ ( − T , ⋅ ) , b + 1 } ] ≤ ( b + 1 ) S t + T [ min { λ ∗ ( b + 1 ) b ∣ ln b ∣ H μ ( − T , ⋅ ) , 1 } ] → 0 as T → ∞ T\to\infty T → ∞ (66) with η = μ \eta=\mu η = μ b H μ = H b μ bH_\mu=H_{b\mu} b H μ = H b μ (114) and interior parabolic estimates give
S t + T [ b g ( H μ ( − T , ⋅ ) ) ] → u b μ ( t , ⋅ ) in C l o c 1 , 2 ( R × R N ) as T → ∞ . (119) S_{t+T}[bg(H_\mu(-T,\cdot))]\to u_{b\mu}(t,\cdot)\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N)\text{ as }T\to\infty. \tag*{(119)} S t + T [ b g ( H μ ( − T , ⋅ ))] → u b μ ( t , ⋅ ) in C loc 1 , 2 ( R × R N ) as T → ∞. (119) For b = 1 b=1 b = 1 (114) .
For 0 < b < 1 0<b<1 0 < b < 1 g g g
b S t + T [ g ( H μ ( − T , ⋅ ) ) ] ≤ S t + T [ b g ( H μ ( − T , ⋅ ) ) ] ≤ S t + T [ g ( b H μ ( − T , ⋅ ) ) ] . bS_{t+T}[g(H_\mu(-T,\cdot))]\leq S_{t+T}[bg(H_\mu(-T,\cdot))]\leq S_{t+T}[g(bH_\mu(-T,\cdot))]. b S t + T [ g ( H μ ( − T , ⋅ ))] ≤ S t + T [ b g ( H μ ( − T , ⋅ ))] ≤ S t + T [ g ( b H μ ( − T , ⋅ ))] . Letting T → ∞ T \to\infty T → ∞ b u μ ≤ u b μ b u_\mu\leq u_{b\mu} b u μ ≤ u b μ
S t + T [ b u μ ( − T , ⋅ ) ] ≤ u b μ ( t , ⋅ ) , S_{t+T}[b u_\mu(-T,\cdot)] \leq u_{b\mu}(t,\cdot), S t + T [ b u μ ( − T , ⋅ )] ≤ u b μ ( t , ⋅ ) , and hence D b u μ ≤ u b μ D_bu_\mu\leq u_{b\mu} D b u μ ≤ u b μ u μ ≥ g ( H μ ) u_\mu\geq g(H_\mu) u μ ≥ g ( H μ )
S t + T [ b u μ ( − T , ⋅ ) ] ≥ S t + T [ b g ( H μ ( − T , ⋅ ) ) ] . S_{t+T}[b u_\mu(-T,\cdot)] \geq S_{t+T}[b g(H_\mu(-T,\cdot))]. S t + T [ b u μ ( − T , ⋅ )] ≥ S t + T [ b g ( H μ ( − T , ⋅ ))] . Letting T → ∞ T \to\infty T → ∞ D b D_b D b (119) give D b u μ ≥ u b μ D_bu_\mu\geq u_{b\mu} D b u μ ≥ u b μ D b u μ = u b μ D_bu_\mu= u_{b\mu} D b u μ = u b μ b > 1 b>1 b > 1 1 / b 1/b 1/ b b μ b\mu b μ D 1 / b u b μ = u ( 1 / b ) ( b μ ) = u μ D_{1/b}u_{b\mu}=u_{(1/b)(b\mu)}=u_\mu D 1/ b u b μ = u ( 1/ b ) ( b μ ) = u μ D b D_b D b D b u μ = u b μ D_bu_\mu=u_{b\mu} D b u μ = u b μ (115) , this proves (113) .
The critical trace and its nonvanishing
We extract a measure on the critical sphere from the small-amplitude limit and prove that this measure is nonzero for each nonzero solution satisfying the critical time–dilation identity. A spherical averaging inequality connects the trace to the radial solution constructed above.
Throughout this section the reaction satisfies (2) , with f ′ ( 0 ) > 0 f'(0)>0 f ′ ( 0 ) > 0 U t = Δ U + f ( U ) U_t=\Delta U+f(U) U t = Δ U + f ( U ) ϕ ∗ \phi_* ϕ ∗ c ∗ c_* c ∗
ϕ ∗ ( s ) ∼ ( − s ) e λ ∗ s ( s → − ∞ ) . \phi_*(s)\sim(-s)e^{\lambda_*s}\quad(s\to-\infty). ϕ ∗ ( s ) ∼ ( − s ) e λ ∗ s ( s → − ∞ ) . For a finite nonnegative measure μ \mu μ S N − 1 \mathbb{S}^{N-1} S N − 1
L μ ( x ) = ∫ S N − 1 e λ ∗ x ⋅ ν μ ( d ν ) , H μ ( t , x ) = e 2 f ′ ( 0 ) t L μ ( x ) . L_\mu(x)=\int_{\mathbb{S}^{N-1}}e^{\lambda_*x\cdot\nu}\,\mu(d\nu),\qquad H_\mu(t,x)=e^{2f'(0)t}L_\mu(x). L μ ( x ) = ∫ S N − 1 e λ ∗ x ⋅ ν μ ( d ν ) , H μ ( t , x ) = e 2 f ′ ( 0 ) t L μ ( x ) . For μ ≠ 0 \mu\ne0 μ = 0 z μ = λ ∗ − 1 ln H μ z_\mu=\lambda_*^{-1}\ln H_\mu z μ = λ ∗ − 1 ln H μ σ \sigma σ ℓ ( R ) = L σ ( R e 1 ) \ell(R)=L_\sigma(Re_1) ℓ ( R ) = L σ ( R e 1 ) N = 1 N=1 N = 1 σ = ( δ − 1 + δ 1 ) / 2 \sigma=(\delta_{-1}+\delta_1)/2 σ = ( δ − 1 + δ 1 ) /2 ℓ ( R ) = cosh ( λ ∗ R ) \ell(R)=\cosh(\lambda_*R) ℓ ( R ) = cosh ( λ ∗ R )
The preceding construction gives the minimal entire solution u μ u_\mu u μ ϕ ∗ ( z μ ) \phi_*(z_\mu) ϕ ∗ ( z μ ) U r a d = u σ U^{\rm rad}=u_\sigma U rad = u σ
ϕ ∗ ( z μ ( t , x ) ) ≤ u μ ( t , x ) ≤ U r a d ( z μ ( t , x ) / c ∗ , 0 ) , \phi_*(z_\mu(t,x))\leq u_\mu(t,x)\leq U^{\rm rad}(z_\mu(t,x)/c_*,0), ϕ ∗ ( z μ ( t , x )) ≤ u μ ( t , x ) ≤ U rad ( z μ ( t , x ) / c ∗ , 0 ) , D b u μ = u b μ , u b μ ( t , x ) = u μ ( t + ln b 2 f ′ ( 0 ) , x ) , U r a d ( t , 0 ) → 0 ( t → − ∞ ) . D_bu_\mu=u_{b\mu},\qquad u_{b\mu}(t,x)=u_\mu\left(t+\frac{\ln b}{2f'(0)},x\right),\qquad U^{\rm rad}(t,0)\to0\quad(t\to-\infty). D b u μ = u b μ , u b μ ( t , x ) = u μ ( t + 2 f ′ ( 0 ) ln b , x ) , U rad ( t , 0 ) → 0 ( t → − ∞ ) . We also use the finite critical-field estimate already proved: for every finite angular measure η \eta η C > 0 C>0 C > 0 ( t , x ) (t,x) ( t , x )
lim T → ∞ ( S t + T [ min { C e − 2 f ′ ( 0 ) T L η , 1 } ] ) ( x ) = 0 , (120) \lim_{T\to\infty}\left(S_{t+T}\left[\min\{Ce^{-2f'(0)T}L_\eta,1\}\right]\right)(x)=0, \tag*{(120)} T → ∞ lim ( S t + T [ min { C e − 2 f ′ ( 0 ) T L η , 1 } ] ) ( x ) = 0 , (120) with local C 1 , 2 C^{1,2} C 1 , 2 [ 0 , 1 ] [0,1] [ 0 , 1 ]
The ordered vector space constructed above has D a D_a D a ⊕ \oplus ⊕ U U U (1) with 0 < U < 1 0<U<1 0 < U < 1 J : E U → C ( K U ) J:E_U\to C(K_U) J : E U → C ( K U ) J ( U ) = 1 J(U)=1 J ( U ) = 1 e λ ∗ h ⋅ ξ + s λ e^{\lambda_*h\cdot\xi+s\lambda} e λ ∗ h ⋅ ξ + s λ λ = f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) \lambda=f'(0)(1+|\xi|^2) λ = f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) ∣ ξ ∣ ≤ 1 |\xi|\leq1 ∣ ξ ∣ ≤ 1
A critical solution U ∗ U_* U ∗ (1) with 0 ≤ U ∗ < 1 0\leq U_*<1 0 ≤ U ∗ < 1
D a U ∗ ( t , x ) = U ∗ ( t + ln a 2 f ′ ( 0 ) , x ) for a > 0. (121) D_aU_*(t,x)=U_*\left(t+\frac{\ln a}{2f'(0)},x\right)\qquad\text{for }a>0. \tag*{(121)} D a U ∗ ( t , x ) = U ∗ ( t + 2 f ′ ( 0 ) ln a , x ) for a > 0. (121) We first obtain a critical-sphere measure from the small-amplitude limit of an entire solution. This limit exists for every 0 < U < 1 0<U<1 0 < U < 1 (121) . In the following proof, F 1 ( u ) = f ′ ( 0 ) u − f ( u ) F_1(u)=f'(0)u-f(u) F 1 ( u ) = f ′ ( 0 ) u − f ( u ) M M M (60) .
Theorem 5.3. For every entire solution U U U 0 < U < 1 0<U<1 0 < U < 1 μ ∗ \mu_* μ ∗ S N − 1 \mathbb{S}^{N-1} S N − 1
D a U ( t , x ) a ln ( 1 / a ) → H ∗ [ U ] ( t , x ) = e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ x ⋅ ν μ ∗ ( d ν ) in C l o c 1 , 2 ( R × R N ) as a ↘ 0. (122) \frac{D_aU(t,x)}{a\ln(1/a)} \to H_*[U](t,x)=e^{2f'(0)t}\int_{\mathbb{S}^{N-1}}e^{\lambda_*x\cdot\nu}\,\mu_*(d\nu) \quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N)\quad\text{as }a\searrow0. \tag*{(122)} a ln ( 1/ a ) D a U ( t , x ) → H ∗ [ U ] ( t , x ) = e 2 f ′ ( 0 ) t ∫ S N − 1 e λ ∗ x ⋅ ν μ ∗ ( d ν ) in C loc 1 , 2 ( R × R N ) as a ↘ 0. (122) The measure may be zero. If it is nonzero, then
U ( t , x ) ≥ ϕ ∗ ( 1 λ ∗ ln ( λ ∗ H ∗ [ U ] ( t , x ) ) ) for all ( t , x ) ∈ R × R N . (123) U(t,x)\geq\phi_*\left(\frac{1}{\lambda_*}\ln(\lambda_*H_*[U](t,x))\right)\quad\text{for all }(t,x)\in\mathbb{R}\times\mathbb{R}^N. \tag*{(123)} U ( t , x ) ≥ ϕ ∗ ( λ ∗ 1 ln ( λ ∗ H ∗ [ U ] ( t , x )) ) for all ( t , x ) ∈ R × R N . (123) Proof. Fix ( t , x ) (t,x) ( t , x ) G ( a ) = D a U ( t , x ) G(a)=D_aU(t,x) G ( a ) = D a U ( t , x ) Lemma 2.19 , and (42) with γ = 1 \gamma=1 γ = 1 Lemma 3.7 , with its quadratic constant M / f ′ ( 0 ) M/f'(0) M / f ′ ( 0 ) f ( G ) / f ′ ( 0 ) ≥ G − ( M / f ′ ( 0 ) ) G 2 f(G)/f'(0)\geq G-(M/f'(0))G^2 f ( G ) / f ′ ( 0 ) ≥ G − ( M / f ′ ( 0 )) G 2 H ∗ [ U ] H_*[U] H ∗ [ U ]
For 0 < a < 1 0<a<1 0 < a < 1 b a = a ln ( 1 / a ) b_a=a\ln(1/a) b a = a ln ( 1/ a ) Z a = D a U / b a Z_a=D_aU/b_a Z a = D a U / b a D a U D_aU D a U
e − 2 f ′ ( 0 ) ∣ t ∣ − λ ∗ ∣ x ∣ Z a ( 0 , 0 ) ≤ Z a ( t , x ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ + λ ∗ ∣ x ∣ Z a ( 0 , 0 ) . e^{-2f'(0)|t|-\lambda_*|x|}Z_a(0,0)\leq Z_a(t,x)\leq e^{2f'(0)|t|+\lambda_*|x|}Z_a(0,0). e − 2 f ′ ( 0 ) ∣ t ∣ − λ ∗ ∣ x ∣ Z a ( 0 , 0 ) ≤ Z a ( t , x ) ≤ e 2 f ′ ( 0 ) ∣ t ∣ + λ ∗ ∣ x ∣ Z a ( 0 , 0 ) . The pointwise limit at ( 0 , 0 ) (0,0) ( 0 , 0 ) Z a Z_a Z a a a a
( Z a ) t − Δ Z a = f ′ ( 0 ) Z a − F 1 ( b a Z a ) b a , 0 ≤ F 1 ( b a Z a ) b a ≤ M b a Z a 2 . (Z_a)_t-\Delta Z_a=f'(0)Z_a-\frac{F_1(b_aZ_a)}{b_a},\qquad0\leq\frac{F_1(b_aZ_a)}{b_a}\leq Mb_aZ_a^2. ( Z a ) t − Δ Z a = f ′ ( 0 ) Z a − b a F 1 ( b a Z a ) , 0 ≤ b a F 1 ( b a Z a ) ≤ M b a Z a 2 . Interior parabolic estimates and the pointwise limit yield convergence in C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2
( H ∗ ) t = Δ H ∗ + f ′ ( 0 ) H ∗ , ∣ ∇ H ∗ ∣ ≤ λ ∗ H ∗ . (H_*)_t=\Delta H_*+f'(0)H_*,\qquad|\nabla H_*|\leq\lambda_*H_*. ( H ∗ ) t = Δ H ∗ + f ′ ( 0 ) H ∗ , ∣∇ H ∗ ∣ ≤ λ ∗ H ∗ . The two-sided comparison shows that H ∗ H_* H ∗ § 7 gives
H ∗ ( t , x ) = ∫ B 1 e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t μ ( d ξ ) . (124) H_*(t,x)=\int_{B_1}e^{\lambda_*x\cdot\xi+f'(0)(1+|\xi|^2)t}\,\mu(d\xi). \tag*{(124)} H ∗ ( t , x ) = ∫ B 1 e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t μ ( d ξ ) . (124) Indeed the representing measure has mass H ∗ ( 0 , 0 ) < ∞ H_*(0,0)<\infty H ∗ ( 0 , 0 ) < ∞ H ∗ ( 0 , s e ) ≤ e λ ∗ s H ∗ ( 0 , 0 ) H_*(0,se)\leq e^{\lambda_*s}H_*(0,0) H ∗ ( 0 , se ) ≤ e λ ∗ s H ∗ ( 0 , 0 ) e e e s ≥ 0 s\geq0 s ≥ 0 e ⋅ ξ > 1 e\cdot\xi>1 e ⋅ ξ > 1 B 1 B_1 B 1
Divide (43) by a ln ( 1 / a ) a\ln(1/a) a ln ( 1/ a ) ϕ ∗ \phi_* ϕ ∗
H ∗ ( t , x ) ≤ 1 λ ∗ exp ( λ ∗ ϕ ∗ − 1 ( U ( t , x ) ) ) , H_*(t,x)\leq\frac{1}{\lambda_*}\exp\bigl(\lambda_*\phi_*^{-1}(U(t,x))\bigr), H ∗ ( t , x ) ≤ λ ∗ 1 exp ( λ ∗ ϕ ∗ − 1 ( U ( t , x )) ) , and hence (123) . In particular, the fixed critical wave ϕ ∗ ( x ⋅ ν + c ∗ t ) \phi_*(x\cdot\nu+c_*t) ϕ ∗ ( x ⋅ ν + c ∗ t ) λ ∗ − 1 δ ν \lambda_*^{-1}\delta_\nu λ ∗ − 1 δ ν
It remains to exclude interior mass in (124) . If μ ( B 1 ) > 0 \mu(B_1)>0 μ ( B 1 ) > 0 r ∈ [ 0 , 1 ) r\in[0,1) r ∈ [ 0 , 1 ) μ ( B ‾ r ) > 0 \mu(\overline{B}_r)>0 μ ( B r ) > 0
h ( t , x ) = ∫ B ‾ r e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t μ ( d ξ ) . h(t,x)=\int_{\overline{B}_r}e^{\lambda_*x\cdot\xi+f'(0)(1+|\xi|^2)t}\,\mu(d\xi). h ( t , x ) = ∫ B r e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t μ ( d ξ ) . Then 0 < h ≤ H ∗ 0<h\leq H_* 0 < h ≤ H ∗ U ≥ ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ h ) ) U\geq\phi_*(\lambda_*^{-1}\ln(\lambda_*h)) U ≥ ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ h )) m > 0 m>0 m > 0 d m > 0 d_m>0 d m > 0
ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ s ) ) ≥ m s for 0 < s ≤ d m . \phi_*(\lambda_*^{-1}\ln(\lambda_*s))\geq ms\quad\text{for }0<s\leq d_m. ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ s )) ≥ m s for 0 < s ≤ d m . Choose 1 < p ≤ 2 1<p\leq2 1 < p ≤ 2 p r 2 < 1 pr^2<1 p r 2 < 1 c = f ′ ( 0 ) ( p − 1 ) ( 1 − p r 2 ) > 0 c=f'(0)(p-1)(1-pr^2)>0 c = f ′ ( 0 ) ( p − 1 ) ( 1 − p r 2 ) > 0 B m B_m B m
B m ≥ max { 1 , M / c } , B m − 1 / ( p − 1 ) / m ≤ d m . B_m\geq\max\{1,M/c\},\qquad B_m^{-1/(p-1)}/m\leq d_m. B m ≥ max { 1 , M / c } , B m − 1/ ( p − 1 ) / m ≤ d m . The lower barrier from the finite linear reconstruction is
ℓ m = ( m h − B m ( m h ) p ) + . \ell_m=(mh-B_m(mh)^p)_+. ℓ m = ( mh − B m ( mh ) p ) + . On its positive set m h < 1 mh<1 mh < 1 h ≤ d m h\le d_m h ≤ d m ℓ m ≤ m h ≤ U \ell_m\le mh\le U ℓ m ≤ mh ≤ U
( ∂ t − Δ − f ′ ( 0 ) ) ( m h ) p ≥ c ( m h ) p , ( ∂ t − Δ ) ℓ m − f ( ℓ m ) ≤ − B m c ( m h ) p + M ( m h ) 2 ≤ 0 (\partial_t-\Delta-f'(0))(mh)^p\ge c(mh)^p,\qquad(\partial_t-\Delta)\ell_m-f(\ell_m)\le-B_m c(mh)^p+M(mh)^2\le0 ( ∂ t − Δ − f ′ ( 0 )) ( mh ) p ≥ c ( mh ) p , ( ∂ t − Δ ) ℓ m − f ( ℓ m ) ≤ − B m c ( mh ) p + M ( mh ) 2 ≤ 0 on that set; taking the maximum with zero preserves the subsolution inequality.
Let V m h f V^f_{mh} V mh f Theorem 3.5 . Since
0 ≤ min { m h , 1 } − ℓ m ≤ B m ( m h ) p , 0\le\min\{mh,1\}-\ell_m\le B_m(mh)^p, 0 ≤ min { mh , 1 } − ℓ m ≤ B m ( mh ) p , comparison, (61) , and (73) give
0 ≤ V m h f ( t , x ) − ( S t + T [ ℓ m ( − T ) ] ) ( x ) ≤ B m e − c ( t + T ) ( m h ( t , x ) ) p → 0 as T → ∞ . 0\le V^f_{mh}(t,x)-\left(S_{t+T}[\ell_m(-T)]\right)(x)
\le B_m e^{-c(t+T)}(mh(t,x))^p\to0\quad\text{as }T\to\infty. 0 ≤ V mh f ( t , x ) − ( S t + T [ ℓ m ( − T )] ) ( x ) ≤ B m e − c ( t + T ) ( mh ( t , x ) ) p → 0 as T → ∞. Because ℓ m ≤ U \ell_m\le U ℓ m ≤ U V m h f ≤ U V^f_{mh}\le U V mh f ≤ U m > 0 m>0 m > 0 s ≥ 0 s\ge0 s ≥ 0
h ( t + s , x ) ≤ e f ′ ( 0 ) ( 1 + r 2 ) s h ( t , x ) , V h f ( t + s , x ) ≤ V m h f ( t , x ) ≤ U ( t , x ) for m ≥ e f ′ ( 0 ) ( 1 + r 2 ) s . h(t+s,x)\le e^{f'(0)(1+r^2)s}h(t,x),\qquad V^f_h(t+s,x)\le V^f_{mh}(t,x)\le U(t,x)\quad\text{for }m\ge e^{f'(0)(1+r^2)s}. h ( t + s , x ) ≤ e f ′ ( 0 ) ( 1 + r 2 ) s h ( t , x ) , V h f ( t + s , x ) ≤ V mh f ( t , x ) ≤ U ( t , x ) for m ≥ e f ′ ( 0 ) ( 1 + r 2 ) s . The first inequality is immediate from the support of μ \mu μ V h f V^f_h V h f (1) with 0 < V h f < 1 0<V^f_h<1 0 < V h f < 1 V h f ( t + s , x ) → 1 V^f_h(t+s,x)\to1 V h f ( t + s , x ) → 1 s → ∞ s\to\infty s → ∞ U < 1 U<1 U < 1 μ ( B 1 ) = 0 \mu(B_1)=0 μ ( B 1 ) = 0 (122) .
For solutions satisfying the time–dilation identity, the amplitude trace becomes a precise past-time asymptotic profile.
Corollary 5.4. Suppose that
D a U ( t , x ) = U ( t + ln a 2 f ′ ( 0 ) , x ) for all a > 0. D_aU(t,x)=U\left(t+\frac{\ln a}{2f'(0)},x\right)\quad\text{for all }a>0. D a U ( t , x ) = U ( t + 2 f ′ ( 0 ) ln a , x ) for all a > 0. Then
Δ U ≥ 1 4 f ′ ( 0 ) U t t , U ( t , ⋅ ) ( − 2 f ′ ( 0 ) t ) e 2 f ′ ( 0 ) t → κ in C l o c 2 ( R N ) as t → − ∞ , \Delta U\ge\frac{1}{4f'(0)}U_{tt},\qquad
\frac{U(t,\cdot)}{(-2f'(0)t)e^{2f'(0)t}}\to\kappa\quad\text{in }C^2_{\mathrm{loc}}(\mathbb{R}^N)\quad\text{as }t\to-\infty, Δ U ≥ 4 f ′ ( 0 ) 1 U tt , ( − 2 f ′ ( 0 ) t ) e 2 f ′ ( 0 ) t U ( t , ⋅ ) → κ in C loc 2 ( R N ) as t → − ∞ , where κ ( x ) = ∫ S N − 1 e λ ∗ x ⋅ ν μ ∗ ( d ν ) \kappa(x)=\int_{S^{N-1}}e^{\lambda^*x\cdot\nu}\,\mu_*(d\nu) κ ( x ) = ∫ S N − 1 e λ ∗ x ⋅ ν μ ∗ ( d ν ) and Δ κ = f ′ ( 0 ) κ \Delta\kappa=f'(0)\kappa Δ κ = f ′ ( 0 ) κ
Proof. The covariance makes G ( a ) = D a U ( t , x ) G(a)=D_aU(t,x) G ( a ) = D a U ( t , x ) a a a a = 1 a=1 a = 1
G ′ ( 1 ) = 1 2 f ′ ( 0 ) U t , G ′ ′ ( 1 ) = 1 4 f ′ ( 0 ) 2 U t t − 1 2 f ′ ( 0 ) U t . G'(1)=\frac{1}{2f'(0)}U_t,\qquad G''(1)=\frac{1}{4f'(0)^2}U_{tt}-\frac{1}{2f'(0)}U_t. G ′ ( 1 ) = 2 f ′ ( 0 ) 1 U t , G ′′ ( 1 ) = 4 f ′ ( 0 ) 2 1 U tt − 2 f ′ ( 0 ) 1 U t . Using (42) with γ = 1 \gamma=1 γ = 1 U t t / ( 4 f ′ ( 0 ) ) ≤ U t − f ( U ) = Δ U U_{tt}/(4f'(0))\le U_t-f(U)=\Delta U U tt / ( 4 f ′ ( 0 )) ≤ U t − f ( U ) = Δ U (122) at reference time zero with a = e 2 f ′ ( 0 ) t a=e^{2f'(0)t} a = e 2 f ′ ( 0 ) t
To prove that the critical trace does not vanish, we first compare the spherical average of a critical solution with its value at a later time. The comparison follows from the translation representation and the nonlinear averaging inequality.
Lemma 5.5. If U ∗ U_* U ∗ is a nonzero critical solution in the sense of (121) , then
∫ S N − 1 U ∗ ( t , R ν ) σ ( d ν ) ≤ U ∗ ( t + ln L σ ( R e 1 ) 2 f ′ ( 0 ) , 0 ) for R ≥ 0 , t ∈ R . (125) \int_{S^{N-1}}U_*(t,R\nu)\,\sigma(d\nu)\le U_*\left(t+\frac{\ln L_\sigma(Re_1)}{2f'(0)},0\right)\quad\text{for }R\ge0,\ t\in\mathbb{R}. \tag*{(125)} ∫ S N − 1 U ∗ ( t , R ν ) σ ( d ν ) ≤ U ∗ ( t + 2 f ′ ( 0 ) ln L σ ( R e 1 ) , 0 ) for R ≥ 0 , t ∈ R . (125) Proof. Work in the principal ideal E U ∗ EU_* E U ∗ J : E U ∗ → C ( K U ∗ ) J:EU_* \to C(K_{U_*}) J : E U ∗ → C ( K U ∗ ) J ( U ∗ ) = 1 J(U_*)=1 J ( U ∗ ) = 1 T s , 0 U ∗ = D e 2 f ′ ( 0 ) s U ∗ T_{s,0}U_* = D_{e^{2f'(0)s}}U_* T s , 0 U ∗ = D e 2 f ′ ( 0 ) s U ∗ J J J e s λ ( k ) = e 2 f ′ ( 0 ) s e^{s\lambda(k)}=e^{2f'(0)s} e s λ ( k ) = e 2 f ′ ( 0 ) s k ∈ K U ∗ k\in K_{U_*} k ∈ K U ∗ λ = 2 f ′ ( 0 ) \lambda=2f'(0) λ = 2 f ′ ( 0 ) ∣ ξ ( k ) ∣ = 1 |\xi(k)|=1 ∣ ξ ( k ) ∣ = 1 K U ∗ K_{U_*} K U ∗
f R ( ζ ) = ∫ S N − 1 e λ ∗ R ν ⋅ ζ σ ( d ν ) for ζ ∈ [ − 1 , 1 ] N . f_R(\zeta)=\int_{S^{N-1}}e^{\lambda_*R\nu\cdot\zeta}\,\sigma(d\nu)\qquad\text{for }\zeta\in[-1,1]^N. f R ( ζ ) = ∫ S N − 1 e λ ∗ R ν ⋅ ζ σ ( d ν ) for ζ ∈ [ − 1 , 1 ] N . Rotational invariance of σ \sigma σ f R ( ξ ( k ) ) = L σ ( R e 1 ) f_R(\xi(k))=L_\sigma(Re_1) f R ( ξ ( k )) = L σ ( R e 1 ) K U ∗ K_{U_*} K U ∗
For each m ≥ 1 m\geq1 m ≥ 1 ν m , 1 , ⋯ , ν m , k m ∈ S N − 1 \nu_{m,1},\cdots,\nu_{m,k_m}\in S^{N-1} ν m , 1 , ⋯ , ν m , k m ∈ S N − 1 S N − 1 ⊂ ⋃ j = 1 k m B 1 / m ( ν m , j ) S^{N-1}\subset\bigcup_{j=1}^{k_m}B_{1/m}(\nu_{m,j}) S N − 1 ⊂ ⋃ j = 1 k m B 1/ m ( ν m , j )
A m , j : = ( B 1 / m ( ν m , j ) ∩ S N − 1 ) ∖ ⋃ i < j B 1 / m ( ν m , i ) for j = 1 , … , k m . A_{m,j}:=\left(B_{1/m}(\nu_{m,j})\cap S^{N-1}\right)\setminus\bigcup_{i<j}B_{1/m}(\nu_{m,i})\qquad\text{for }j=1,\ldots,k_m. A m , j := ( B 1/ m ( ν m , j ) ∩ S N − 1 ) ∖ i < j ⋃ B 1/ m ( ν m , i ) for j = 1 , … , k m . These sets form a Borel partition of the sphere, and ∣ ν − ν m , j ∣ < 1 / m |\nu-\nu_{m,j}|<1/m ∣ ν − ν m , j ∣ < 1/ m ν ∈ A m , j \nu\in A_{m,j} ν ∈ A m , j θ m , j : = σ ( A m , j ) ≥ 0 \theta_{m,j}:=\sigma(A_{m,j})\geq0 θ m , j := σ ( A m , j ) ≥ 0 ∑ j = 1 k m θ m , j = 1 \sum_{j=1}^{k_m}\theta_{m,j}=1 ∑ j = 1 k m θ m , j = 1 θ m , j = 0 \theta_{m,j}=0 θ m , j = 0
f R , m ( ζ ) = ∑ j = 1 k m θ m , j e λ ∗ R ν m , j ⋅ ζ , Q m = ⨁ j = 1 k m D θ m , j ( T 0 , R ν m , j U ∗ ) . f_{R,m}(\zeta)=\sum_{j=1}^{k_m}\theta_{m,j}e^{\lambda_*R\nu_{m,j}\cdot\zeta},\qquad Q_m=\bigoplus_{j=1}^{k_m}D_{\theta_{m,j}}(T_{0,R\nu_{m,j}}U_*). f R , m ( ζ ) = j = 1 ∑ k m θ m , j e λ ∗ R ν m , j ⋅ ζ , Q m = j = 1 ⨁ k m D θ m , j ( T 0 , R ν m , j U ∗ ) . Thus Q m ≥ 0 Q_m\geq0 Q m ≥ 0 f R , m → f R f_{R,m}\to f_R f R , m → f R [ − 1 , 1 ] N [-1,1]^N [ − 1 , 1 ] N m → ∞ m\to\infty m → ∞ J ( Q m ) ( k ) = f R , m ( ξ ( k ) ) J(Q_m)(k)=f_{R,m}(\xi(k)) J ( Q m ) ( k ) = f R , m ( ξ ( k )) J ( D L σ ( R e 1 ) U ∗ ) = L σ ( R e 1 ) J(D_{L_\sigma(Re_1)}U_*)=L_\sigma(Re_1) J ( D L σ ( R e 1 ) U ∗ ) = L σ ( R e 1 )
∥ Q m ⊖ D L σ ( R e 1 ) U ∗ ∥ U ∗ ≤ ∥ f R , m − f R ∥ C ( [ − 1 , 1 ] N ) → 0. \lVert Q_m\ominus D_{L_\sigma(Re_1)}U_*\rVert_{U_*}\leq\lVert f_{R,m}-f_R\rVert_{C([-1,1]^N)}\to0. ∥ Q m ⊖ D L σ ( R e 1 ) U ∗ ∥ U ∗ ≤ ∥ f R , m − f R ∥ C ([ − 1 , 1 ] N ) → 0. The subtraction inside this norm is subtraction in the ordered space. Apply (53) to the finite average Q m Q_m Q m ( t , 0 ) (t,0) ( t , 0 )
Q m ( t , 0 ) ≥ ∑ j = 1 k m θ m , j U ∗ ( t , R ν m , j ) . Q_m(t,0)\geq\sum_{j=1}^{k_m}\theta_{m,j}U_*(t,R\nu_{m,j}). Q m ( t , 0 ) ≥ j = 1 ∑ k m θ m , j U ∗ ( t , R ν m , j ) . The left side tends to D L σ ( R e 1 ) U ∗ ( t , 0 ) D_{L_\sigma(Re_1)}U_*(t,0) D L σ ( R e 1 ) U ∗ ( t , 0 ) m → ∞ m\to\infty m → ∞ Lemma 2.22 , while
∑ j = 1 k m θ m , j U ∗ ( t , R ν m , j ) → ∫ S N − 1 U ∗ ( t , R ν ) σ ( d ν ) as m → ∞ . \sum_{j=1}^{k_m}\theta_{m,j}U_*(t,R\nu_{m,j})\to\int_{S^{N-1}}U_*(t,R\nu)\,\sigma(d\nu)\qquad\text{as }m\to\infty. j = 1 ∑ k m θ m , j U ∗ ( t , R ν m , j ) → ∫ S N − 1 U ∗ ( t , R ν ) σ ( d ν ) as m → ∞. By (121) with a = L σ ( R e 1 ) a=L_\sigma(Re_1) a = L σ ( R e 1 )
D L σ ( R e 1 ) U ∗ ( t , 0 ) = U ∗ ( t + ln L σ ( R e 1 ) 2 f ′ ( 0 ) , 0 ) . D_{L_\sigma(Re_1)}U_*(t,0)=U_*\left(t+\frac{\ln L_\sigma(Re_1)}{2f'(0)},0\right). D L σ ( R e 1 ) U ∗ ( t , 0 ) = U ∗ ( t + 2 f ′ ( 0 ) ln L σ ( R e 1 ) , 0 ) . This proves (125) .
We next use (125) to prove that H_*[U_*]\nequiv0 for every entire solution 0 < U ∗ < 1 0<U_*<1 0 < U ∗ < 1 (121) , where H ∗ [ U ∗ ] H_*[U_*] H ∗ [ U ∗ ] (122) . We use the solutions u r a d = u σ u^{\mathrm{rad}}=u_\sigma u rad = u σ u μ u_\mu u μ Theorems 5.1 and 5.2.
The lower bound from the trace and the radial upper bound now identify every nonzero critical solution with its canonical measure-generated solution.
Theorem 5.6. Let U ∗ U_* U ∗ (121) . Let μ ∗ : = μ ∗ [ U ∗ ] \mu_*:=\mu_*[U_*] μ ∗ := μ ∗ [ U ∗ ] Theorem 5.3 . With L μ ∗ L_{\mu_*} L μ ∗ (12) , we have
H ∗ [ U ∗ ] ( t , x ) = e 2 f ′ ( 0 ) t L μ ∗ ( x ) . H_*[U_*](t,x)=e^{2f'(0)t}L_{\mu_*}(x). H ∗ [ U ∗ ] ( t , x ) = e 2 f ′ ( 0 ) t L μ ∗ ( x ) . Then μ ∗ ≠ 0 \mu_* \ne0 μ ∗ = 0
U ∗ ( t , x ) ≤ U r a d ( t + ln ( λ ∗ L μ ∗ ( x ) ) 2 f ′ ( 0 ) , 0 ) . (126) U_*(t,x) \le U^{\mathrm{rad}}\left(t+\frac{\ln(\lambda_*L_{\mu_*}(x))}{2f'(0)},0\right). \tag*{(126)} U ∗ ( t , x ) ≤ U rad ( t + 2 f ′ ( 0 ) ln ( λ ∗ L μ ∗ ( x )) , 0 ) . (126) In particular, for any entire solution 0 ≤ U ∗ < 1 0 \le U_* < 1 0 ≤ U ∗ < 1 (121) ,
H ∗ [ U ∗ ] ≡ 0 ⇒ U ∗ ≡ 0. H_*[U_*] \equiv0 \quad\Rightarrow\quad U_* \equiv0. H ∗ [ U ∗ ] ≡ 0 ⇒ U ∗ ≡ 0. Proof. Applying Corollary 5.4 to U ∗ U_* U ∗ x = 0 x=0 x = 0
U ∗ ( s , 0 ) ( − 2 f ′ ( 0 ) s ) e 2 f ′ ( 0 ) s ⟶ L μ ∗ ( 0 ) as s → − ∞ . (127) \frac{U_*(s,0)}{(-2f'(0)s)e^{2f'(0)s}} \longrightarrow L_{\mu_*}(0) \quad\text{as } s\to-\infty. \tag*{(127)} ( − 2 f ′ ( 0 ) s ) e 2 f ′ ( 0 ) s U ∗ ( s , 0 ) ⟶ L μ ∗ ( 0 ) as s → − ∞. (127) Fix any number b > λ ∗ L μ ∗ ( 0 ) b>\lambda_*L_{\mu_*}(0) b > λ ∗ L μ ∗ ( 0 ) (6) and (127) imply U ∗ ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) U_*(z/c_*,0)\le\phi_*(z+\lambda_*^{-1}\ln b) U ∗ ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) z z z U ∗ U_* U ∗ C b > 0 C_b>0 C b > 0
U ∗ ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) + C b e λ ∗ z for every z ∈ R . (128) U_*(z/c_*,0)\le\phi_*(z+\lambda_*^{-1}\ln b)+C_be^{\lambda_*z}
\quad\text{for every } z\in\mathbb{R}. \tag*{(128)} U ∗ ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) + C b e λ ∗ z for every z ∈ R . (128) Define the spherical average
W ( t , x ) = ∫ S N − 1 U ∗ ( t , ∣ x ∣ ξ ) d σ ( ξ ) . W(t,x)=\int_{\mathbb{S}^{N-1}}U_*(t,|x|\xi)\,d\sigma(\xi). W ( t , x ) = ∫ S N − 1 U ∗ ( t , ∣ x ∣ ξ ) d σ ( ξ ) . It is a C 1 , 2 C^{1,2} C 1 , 2 W ( t , 0 ) = U ∗ ( t , 0 ) W(t,0)=U_*(t,0) W ( t , 0 ) = U ∗ ( t , 0 ) f f f
W t − Δ W = ∫ S N − 1 f ( U ∗ ( t , ∣ x ∣ ξ ) ) d σ ( ξ ) ≤ f ( W ) , W_t-\Delta W=\int_{\mathbb{S}^{N-1}}f(U_*(t,|x|\xi))\,d\sigma(\xi)\le f(W), W t − Δ W = ∫ S N − 1 f ( U ∗ ( t , ∣ x ∣ ξ )) d σ ( ξ ) ≤ f ( W ) , so W W W (125) , one has
W ( − T , x ) ≤ U ∗ ( − T + ln L σ ( x ) 2 f ′ ( 0 ) , 0 ) . W(-T,x)\le U_*\left(-T+\frac{\ln L_\sigma(x)}{2f'(0)},0\right). W ( − T , x ) ≤ U ∗ ( − T + 2 f ′ ( 0 ) ln L σ ( x ) , 0 ) . Apply (128) with z = − c ∗ T + λ ∗ − 1 ln L σ ( x ) z=-c_*T+\lambda_*^{-1}\ln L_\sigma(x) z = − c ∗ T + λ ∗ − 1 ln L σ ( x ) W ≤ 1 W\le1 W ≤ 1
W ( − T , x ) ≤ ϕ ∗ ( − c ∗ T + λ ∗ − 1 ln ( b L σ ( x ) ) ) + min { C b e − 2 f ′ ( 0 ) T L σ ( x ) , 1 } . W(-T,x)\le\phi_*(-c_*T+\lambda_*^{-1}\ln(bL_\sigma(x)))+\min\{C_be^{-2f'(0)T}L_\sigma(x),1\}. W ( − T , x ) ≤ ϕ ∗ ( − c ∗ T + λ ∗ − 1 ln ( b L σ ( x ))) + min { C b e − 2 f ′ ( 0 ) T L σ ( x ) , 1 } . The comparison principle and semigroup subadditivity give at any fixed ( t , 0 ) (t,0) ( t , 0 ) t > − T t>-T t > − T
U ∗ ( t , 0 ) = W ( t , 0 ) ≤ ( S t + T [ ϕ ∗ ( z b σ ( − T , ⋅ ) ) ] ) ( 0 ) + ( S t + T [ min { C b e − 2 f ′ ( 0 ) T L σ , 1 } ] ) ( 0 ) . U_*(t,0)=W(t,0)\le\left(S_{t+T}[\phi_*(z_{b\sigma}(-T,\cdot))]\right)(0)
+\left(S_{t+T}[\min\{C_be^{-2f'(0)T}L_\sigma,1\}]\right)(0). U ∗ ( t , 0 ) = W ( t , 0 ) ≤ ( S t + T [ ϕ ∗ ( z bσ ( − T , ⋅ ))] ) ( 0 ) + ( S t + T [ min { C b e − 2 f ′ ( 0 ) T L σ , 1 }] ) ( 0 ) . Let T → ∞ T\to\infty T → ∞ u b σ ( t , 0 ) u_{b\sigma}(t,0) u bσ ( t , 0 ) (66) with μ = σ \mu=\sigma μ = σ
U ∗ ( t , 0 ) ≤ u b σ ( t , 0 ) = U r a d ( t + ln b 2 f ′ ( 0 ) , 0 ) for every b > λ ∗ L μ ∗ ( 0 ) . U_*(t,0)\le u_{b\sigma}(t,0)=U^{\mathrm{rad}}\left(t+\frac{\ln b}{2f'(0)},0\right)
\quad\text{for every } b>\lambda_*L_{\mu_*}(0). U ∗ ( t , 0 ) ≤ u bσ ( t , 0 ) = U rad ( t + 2 f ′ ( 0 ) ln b , 0 ) for every b > λ ∗ L μ ∗ ( 0 ) . If L μ ∗ ( 0 ) = 0 L_{\mu_*}(0)=0 L μ ∗ ( 0 ) = 0 b ↘ 0 b\searrow0 b ↘ 0 U ∗ ( t , 0 ) > 0 U_*(t,0)>0 U ∗ ( t , 0 ) > 0 U r a d ( s , 0 ) → 0 U^{\mathrm{rad}}(s,0)\to0 U rad ( s , 0 ) → 0 s → − ∞ s\to-\infty s → − ∞ L μ ∗ ( 0 ) > 0 L_{\mu_*}(0)>0 L μ ∗ ( 0 ) > 0 b ↘ λ ∗ L μ ∗ ( 0 ) b\searrow\lambda_*L_{\mu_*}(0) b ↘ λ ∗ L μ ∗ ( 0 ) (126) at x = 0 x=0 x = 0 U ∗ x ( t , y ) = U ∗ ( t , x + y ) U_*^x(t,y)=U_*(t,x+y) U ∗ x ( t , y ) = U ∗ ( t , x + y ) (126) at every ( t , x ) (t,x) ( t , x )
Identification and weak continuity
We use the critical trace to identify each critical solution with the canonical solution associated with its measure, and then prove weak continuity of this family. The argument starts by determining the exact coefficient in the past asymptotics of the radial solution.
We next determine the precise asymptotic behaviour of U r a d ( t , 0 ) U^{\mathrm{rad}}(t,0) U rad ( t , 0 ) t → − ∞ t\to-\infty t → − ∞
U r a d ( s , 0 ) ∼ ( − c ∗ s ) e 2 f ′ ( 0 ) s for s → − ∞ . (129) U^{\mathrm{rad}}(s,0)\sim(-c_*s)e^{2f'(0)s}\qquad\text{for }s\to-\infty. \tag*{(129)} U rad ( s , 0 ) ∼ ( − c ∗ s ) e 2 f ′ ( 0 ) s for s → − ∞. (129) Let μ ∗ : = μ ∗ [ u σ ] \mu_*:=\mu_*[u_\sigma] μ ∗ := μ ∗ [ u σ ] Theorem 5.3 , so that
lim a ↘ 0 D a u σ ( t , x ) a ln ( 1 / a ) = H ∗ [ u σ ] ( t , x ) = e 2 f ′ ( 0 ) t L μ ∗ ( x ) . \lim_{a\searrow0}\frac{D_a u_\sigma(t,x)}{a\ln(1/a)}=H_*[u_\sigma](t,x)=e^{2f'(0)t}L_{\mu_*}(x). a ↘ 0 lim a ln ( 1/ a ) D a u σ ( t , x ) = H ∗ [ u σ ] ( t , x ) = e 2 f ′ ( 0 ) t L μ ∗ ( x ) . The radial symmetry of the solution passes to its limit. Taking the spherical average and applying (116) with μ = μ ∗ \mu=\mu_* μ = μ ∗
L μ ∗ ( x ) = ∫ S N − 1 L μ ∗ ( ∣ x ∣ ξ ) d σ ( ξ ) = μ ∗ ( S N − 1 ) L σ ( x ) = L μ ∗ ( 0 ) L σ ( x ) . L_{\mu_*}(x)=\int_{\mathbb{S}^{N-1}}L_{\mu_*}(|x|\xi)\,d\sigma(\xi)=\mu_*(\mathbb{S}^{N-1})L_\sigma(x)=L_{\mu_*}(0)L_\sigma(x). L μ ∗ ( x ) = ∫ S N − 1 L μ ∗ ( ∣ x ∣ ξ ) d σ ( ξ ) = μ ∗ ( S N − 1 ) L σ ( x ) = L μ ∗ ( 0 ) L σ ( x ) . By (113) , D a u σ ( 0 , 0 ) = u σ ( ln a 2 f ′ ( 0 ) , 0 ) D_a u_\sigma(0,0)=u_\sigma\left(\frac{\ln a}{2f'(0)},0\right) D a u σ ( 0 , 0 ) = u σ ( 2 f ′ ( 0 ) l n a , 0 )
lim s → − ∞ u σ ( s , 0 ) ( − 2 f ′ ( 0 ) s ) e 2 f ′ ( 0 ) s = L μ ∗ ( 0 ) . \lim_{s\to-\infty}\frac{u_\sigma(s,0)}{(-2f'(0)s)e^{2f'(0)s}}=L_{\mu_*}(0). s → − ∞ lim ( − 2 f ′ ( 0 ) s ) e 2 f ′ ( 0 ) s u σ ( s , 0 ) = L μ ∗ ( 0 ) . Since L σ ( 0 ) = 1 L_\sigma(0)=1 L σ ( 0 ) = 1 Theorem 5.1 gives u σ ( s , 0 ) ≥ ϕ ∗ ( c ∗ s ) u_\sigma(s,0)\geq\phi_*(c_*s) u σ ( s , 0 ) ≥ ϕ ∗ ( c ∗ s ) L μ ∗ ( 0 ) ≥ 1 / λ ∗ L_{\mu_*}(0)\geq1/\lambda_* L μ ∗ ( 0 ) ≥ 1/ λ ∗ (123) and L μ ∗ ( x ) = L μ ∗ ( 0 ) L σ ( x ) L_{\mu_*}(x)=L_{\mu_*}(0)L_\sigma(x) L μ ∗ ( x ) = L μ ∗ ( 0 ) L σ ( x )
u σ ( t , x ) ≥ ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ H ∗ [ u σ ] ( t , x ) ) ) = ϕ ∗ ( z λ ∗ L μ ∗ ( 0 ) σ ( t , x ) ) . u_\sigma(t,x)\geq\phi_*\left(\lambda_*^{-1}\ln\left(\lambda_*H_*[u_\sigma](t,x)\right)\right)=\phi_*\left(z_{\lambda_*L_{\mu_*}(0)\sigma}(t,x)\right). u σ ( t , x ) ≥ ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ H ∗ [ u σ ] ( t , x ) ) ) = ϕ ∗ ( z λ ∗ L μ ∗ ( 0 ) σ ( t , x ) ) . If L μ ∗ ( 0 ) > 1 / λ ∗ L_{\mu_*}(0)>1/\lambda_* L μ ∗ ( 0 ) > 1/ λ ∗ (110) therefore give
u σ ≥ u λ ∗ L μ ∗ ( 0 ) σ = u σ ( t + ln ( λ ∗ L μ ∗ ( 0 ) ) 2 f ′ ( 0 ) , x ) > u σ , u_\sigma\geq u_{\lambda_*L_{\mu_*}(0)\sigma}=u_\sigma\left(t+\frac{\ln(\lambda_*L_{\mu_*}(0))}{2f'(0)},x\right)>u_\sigma, u σ ≥ u λ ∗ L μ ∗ ( 0 ) σ = u σ ( t + 2 f ′ ( 0 ) ln ( λ ∗ L μ ∗ ( 0 )) , x ) > u σ , contradicting strict time monotonicity. Thus L μ ∗ ( 0 ) = 1 / λ ∗ L_{\mu_*}(0)=1/\lambda_* L μ ∗ ( 0 ) = 1/ λ ∗
With the radial tail coefficient fixed, the lower critical profile and the radial upper bound determine the same entire solution. The following uniqueness statement will identify the critical component.
Lemma 5.7. Let μ \mu μ S N − 1 \mathbb{S}^{N-1} S N − 1 Z Z Z (1) satisfying
ϕ ∗ ( z μ ( t , x ) ) ≤ Z ( t , x ) ≤ U r a d ( z μ ( t , x ) / c ∗ , 0 ) . \phi_*(z_\mu(t,x))\leq Z(t,x)\leq U^{\mathrm{rad}}(z_\mu(t,x)/c_*,0). ϕ ∗ ( z μ ( t , x )) ≤ Z ( t , x ) ≤ U rad ( z μ ( t , x ) / c ∗ , 0 ) . Then Z = u μ Z=u_\mu Z = u μ
Proof. Minimality of the canonical entire solution gives Z ≥ u μ Z\geq u_\mu Z ≥ u μ b > 1 b>1 b > 1 (129) , there is C b < ∞ C_b<\infty C b < ∞ z ∈ R z\in\mathbb{R} z ∈ R
U r a d ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) + C b e λ ∗ z . U^{\mathrm{rad}}(z/c_*,0)\leq\phi_*(z+\lambda_*^{-1}\ln b)+C_b e^{\lambda_*z}. U rad ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) + C b e λ ∗ z . Indeed, (129) and (6) give
lim z → − ∞ ϕ ∗ ( z + λ ∗ − 1 ln b ) U r a d ( z / c ∗ , 0 ) = b > 1. (130) \lim_{z\to-\infty}\frac{\phi_*(z+\lambda_*^{-1}\ln b)}{U^{\mathrm{rad}}(z/c_*,0)}=b>1. \tag*{(130)} z → − ∞ lim U rad ( z / c ∗ , 0 ) ϕ ∗ ( z + λ ∗ − 1 ln b ) = b > 1. (130) Thus there exists z b ∈ R z_b \in\mathbb{R} z b ∈ R U rad ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) U^{\operatorname{rad}}(z/c_*,0) \le\phi_*(z+\lambda_*^{-1}\ln b) U rad ( z / c ∗ , 0 ) ≤ ϕ ∗ ( z + λ ∗ − 1 ln b ) z ≤ z b z \le z_b z ≤ z b z ≥ z b z \ge z_b z ≥ z b C b = e − λ ∗ z b C_b=e^{-\lambda_*z_b} C b = e − λ ∗ z b U rad ( z / c ∗ , 0 ) ≤ 1 ≤ C b e λ ∗ z U^{\operatorname{rad}}(z/c_*,0) \le1 \le C_be^{\lambda_*z} U rad ( z / c ∗ , 0 ) ≤ 1 ≤ C b e λ ∗ z Z ≤ 1 Z \le1 Z ≤ 1
Z ( − T , x ) ≤ ϕ ∗ ( z b μ ( − T , x ) ) + min { C b e − 2 f ′ ( 0 ) T L μ ( x ) , 1 } . Z(-T,x) \le\phi_*(z_{b\mu}(-T,x))+\min\{C_be^{-2f'(0)T}L_\mu(x),1\}. Z ( − T , x ) ≤ ϕ ∗ ( z b μ ( − T , x )) + min { C b e − 2 f ′ ( 0 ) T L μ ( x ) , 1 } . Applying the comparison principle and semigroup subadditivity, letting T → ∞ T \to\infty T → ∞ (66) give Z ≤ u b μ Z \le u_{b\mu} Z ≤ u b μ b ↘ 1 b \searrow1 b ↘ 1 (113) and continuity gives Z ≤ u μ Z \le u_\mu Z ≤ u μ
Theorem 5.8. Every nonzero critical solution U ∗ U_* U ∗ (121) has the form
U ∗ = u μ , μ = λ ∗ μ ∗ [ U ∗ ] , U_* = u_\mu,\qquad\mu=\lambda_*\mu_*[U_*], U ∗ = u μ , μ = λ ∗ μ ∗ [ U ∗ ] , where μ \mu μ S N − 1 \mathbb{S}^{N-1} S N − 1
Proof. Theorem 5.6 gives μ ∗ [ U ∗ ] ≠ 0 \mu_*[U_*] \ne0 μ ∗ [ U ∗ ] = 0
U ∗ ≥ ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ H ∗ [ U ∗ ] ) ) = ϕ ∗ ( z μ ) . U_* \ge\phi_*\left(\lambda_*^{-1}\ln(\lambda_*H_*[U_*])\right)=\phi_*(z_\mu). U ∗ ≥ ϕ ∗ ( λ ∗ − 1 ln ( λ ∗ H ∗ [ U ∗ ]) ) = ϕ ∗ ( z μ ) . The universal radial upper bound (126) reads U ∗ ≤ U rad ( z μ / c ∗ , 0 ) U_* \le U^{\operatorname{rad}}(z_\mu/c_*,0) U ∗ ≤ U rad ( z μ / c ∗ , 0 ) U ∗ = u μ U_*=u_\mu U ∗ = u μ
This identification also gives continuity under weak convergence of measures, allowing approximation by finite atomic measures.
Proposition 5.9. If finite nonnegative measures μ n \mu_n μ n S N − 1 \mathbb{S}^{N-1} S N − 1 μ \mu μ
u μ n → u μ in C l o c 1 , 2 ( R × R N ) . u_{\mu_n}\to u_\mu\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N). u μ n → u μ in C loc 1 , 2 ( R × R N ) . Here u 0 = 0 u_0=0 u 0 = 0 U ∗ U_* U ∗ C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2
Proof. Weak convergence on the compact sphere gives
sup n μ n ( S N − 1 ) < ∞ , L μ n → L μ in C l o c ∞ ( R N ) . \sup_n\mu_n(\mathbb{S}^{N-1})<\infty,\qquad L_{\mu_n}\to L_\mu\quad\text{in }C_{\mathrm{loc}}^\infty(\mathbb{R}^N). n sup μ n ( S N − 1 ) < ∞ , L μ n → L μ in C loc ∞ ( R N ) . If μ = 0 \mu=0 μ = 0 R > 0 R>0 R > 0 n n n μ n ≠ 0 \mu_n\ne0 μ n = 0
sup ∣ t ∣ ≤ R ∣ x ∣ ≤ R u μ n ( t , x ) ≤ U rad ( R + ln ( e λ ∗ R μ n ( S N − 1 ) ) 2 f ′ ( 0 ) , 0 ) → 0. \sup_{\substack{|t|\le R\\|x|\le R}}u_{\mu_n}(t,x)\le U^{\operatorname{rad}}\left(R+\frac{\ln(e^{\lambda_*R}\mu_n(\mathbb{S}^{N-1}))}{2f'(0)},0\right)\to0. ∣ t ∣ ≤ R ∣ x ∣ ≤ R sup u μ n ( t , x ) ≤ U rad ( R + 2 f ′ ( 0 ) ln ( e λ ∗ R μ n ( S N − 1 )) , 0 ) → 0. Here we used (112) and L μ n ( x ) ≤ e λ ∗ R μ n ( S N − 1 ) L_{\mu_n}(x)\le e^{\lambda_*R}\mu_n(\mathbb{S}^{N-1}) L μ n ( x ) ≤ e λ ∗ R μ n ( S N − 1 ) ∣ x ∣ ≤ R |x|\le R ∣ x ∣ ≤ R u 0 = 0 u_0=0 u 0 = 0
If μ ≠ 0 \mu\ne0 μ = 0 L μ > 0 L_\mu>0 L μ > 0
z μ n → z μ in C l o c ∞ ( R × R N ) . z_{\mu_n}\to z_\mu\quad\text{in }C_{\mathrm{loc}}^\infty(\mathbb{R}\times\mathbb{R}^N). z μ n → z μ in C loc ∞ ( R × R N ) . By parabolic compactness, one has, up to extraction of a subsequence,
u μ n j → Z in C l o c 1 , 2 ( R × R N ) , ϕ ∗ ( z μ ) ≤ Z ≤ U rad ( z μ / c ∗ , 0 ) . u_{\mu_{n_j}}\to Z\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N),\qquad\phi_*(z_\mu)\le Z\le U^{\operatorname{rad}}(z_\mu/c_*,0). u μ n j → Z in C loc 1 , 2 ( R × R N ) , ϕ ∗ ( z μ ) ≤ Z ≤ U rad ( z μ / c ∗ , 0 ) . Lemma 5.7 gives Z = u μ Z=u_\mu Z = u μ
For U ∗ ≠ 0 U_*\ne0 U ∗ = 0 U ∗ = u μ U_*=u_\mu U ∗ = u μ
μ k = ∑ i = 1 m k b k , i δ ν k , i ⇀ μ , b k , i > 0 , ν k , i ∈ S N − 1 . \mu_k=\sum_{i=1}^{m_k}b_{k,i}\delta_{\nu_{k,i}}\rightharpoonup\mu,\qquad b_{k,i}>0,\quad\nu_{k,i}\in\mathbb{S}^{N-1}. μ k = i = 1 ∑ m k b k , i δ ν k , i ⇀ μ , b k , i > 0 , ν k , i ∈ S N − 1 . Then u μ k → U ∗ u_{\mu_k}\to U_* u μ k → U ∗ C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2 U ∗ = 0 U_*=0 U ∗ = 0 μ k = k − 1 δ ν \mu_k=k^{-1}\delta_\nu μ k = k − 1 δ ν ν ∈ S N − 1 \nu\in\mathbb{S}^{N-1} ν ∈ S N − 1
Proof of Theorem 1.2. Existence and (13) follow from Theorem 5.2. Theorem 2.1 gives strict time monotonicity and the limits at t = ± ∞ t = \pm\infty t = ± ∞ (14) . For α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) s α s_\alpha s α U r a d ( s α , 0 ) = α U^{\mathrm{rad}}(s_\alpha,0)=\alpha U rad ( s α , 0 ) = α (112) imply
u μ ( t , x ) = α ⇒ c ∗ s α ≤ z μ ( t , x ) ≤ ϕ ∗ − 1 ( α ) . u_\mu(t,x)=\alpha\quad\Rightarrow\quad c_*s_\alpha\leq z_\mu(t,x)\leq\phi_*^{-1}(\alpha). u μ ( t , x ) = α ⇒ c ∗ s α ≤ z μ ( t , x ) ≤ ϕ ∗ − 1 ( α ) . Taking C α = 1 + max { ∣ c ∗ s α ∣ , ∣ ϕ ∗ − 1 ( α ) ∣ } C_\alpha=1+\max\{|c_*s_\alpha|,|\phi_*^{-1}(\alpha)|\} C α = 1 + max { ∣ c ∗ s α ∣ , ∣ ϕ ∗ − 1 ( α ) ∣ } (15) .
By (5) , (129) , and the same two bounds,
u μ ( t , x ) ( − c ∗ t ) e 2 f ′ ( 0 ) t → L μ ( x ) as t → − ∞ for every x ∈ R N . \frac{u_\mu(t,x)}{(-c_*t)e^{2f'(0)t}}\to L_\mu(x)\qquad\text{as }t\to-\infty\text{ for every }x\in\mathbb{R}^{N}. ( − c ∗ t ) e 2 f ′ ( 0 ) t u μ ( t , x ) → L μ ( x ) as t → − ∞ for every x ∈ R N . Thus u μ = u ν u_\mu=u_\nu u μ = u ν L μ = L ν L_\mu=L_\nu L μ = L ν μ = ν \mu=\nu μ = ν
The Hamel–Nadirashvili closure theorem
We combine the decomposition and the critical measure identification to prove Theorem 1.1 . We first control mixed approximation at a fixed strict gradient bound, identify the finite atomic constructions as HN solutions, and then choose a diagonal sequence converging to the original entire solution.
The reaction remains the fixed function satisfying (2) , with f ′ ( 0 ) > 0 f'(0)>0 f ′ ( 0 ) > 0 S τ S_\tau S τ L τ = e f ′ ( 0 ) τ P τ L_\tau=e^{f'(0)\tau}P_\tau L τ = e f ′ ( 0 ) τ P τ
Q R = [ − R , R ] × B R ( 0 ) ‾ , ∥ v ∥ 2 , 1 ; R = sup Q R ( ∣ v ∣ + ∣ v t ∣ + ∑ i ∣ v x i ∣ + ∑ i , j ∣ v x i x j ∣ ) . Q_R=[-R,R]\times\overline{B_R(0)},\qquad\|v\|_{2,1;R}=\sup_{Q_R}\left(|v|+|v_t|+\sum_i|v_{x_i}|+\sum_{i,j}|v_{x_ix_j}|\right). Q R = [ − R , R ] × B R ( 0 ) , ∥ v ∥ 2 , 1 ; R = Q R sup ( ∣ v ∣ + ∣ v t ∣ + i ∑ ∣ v x i ∣ + i , j ∑ ∣ v x i x j ∣ ) . Mixed approximation at a fixed spectral radius
For a fixed r < 1 r<1 r < 1 B r B_r B r
For a nonnegative finite measure η \eta η S N − 1 S^{N-1} S N − 1 u η u_\eta u η η ≠ 0 \eta\ne0 η = 0 u 0 = 0 u_0=0 u 0 = 0 L η L_\eta L η z η z_\eta z η (12) and the construction (114) . Theorem 5.2 and the ancient estimates give
u η ( t , x ) ≤ U r a d ( t + ln L η ( x ) 2 f ′ ( 0 ) , 0 ) . (131) u_\eta(t,x)\leq U^{\mathrm{rad}}\left(t+\frac{\ln L_\eta(x)}{2f'(0)},0\right). \tag*{(131)} u η ( t , x ) ≤ U rad ( t + 2 f ′ ( 0 ) ln L η ( x ) , 0 ) . (131) D a u η ( t , x ) = u η ( t + ln a 2 f ′ ( 0 ) , x ) , ∣ ∇ ln u η ∣ ≤ λ ∗ . (132) D_a u_\eta(t,x)=u_\eta\left(t+\frac{\ln a}{2f'(0)},x\right),\qquad|\nabla\ln u_\eta|\leq\lambda_*. \tag*{(132)} D a u η ( t , x ) = u η ( t + 2 f ′ ( 0 ) ln a , x ) , ∣∇ ln u η ∣ ≤ λ ∗ . (132) For η ≠ 0 \eta\ne0 η = 0 ζ \zeta ζ B r ‾ \overline{B_r} B r 0 ≤ r < 1 0\leq r<1 0 ≤ r < 1
H ζ ( t , x ) = ∫ B r ‾ e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t ζ ( d ξ ) . H_\zeta(t,x)=\int_{\overline{B_r}}e^{\lambda_*x\cdot\xi+f'(0)(1+|\xi|^2)t}\,\zeta(d\xi). H ζ ( t , x ) = ∫ B r e λ ∗ x ⋅ ξ + f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) t ζ ( d ξ ) . Let V H ζ V_{H_\zeta} V H ζ Theorem 3.5 , with V H 0 = 0 V_{H_0}=0 V H 0 = 0 0 ≤ V H ζ ≤ min { H ζ , 1 } 0\leq V_{H_\zeta}\leq\min\{H_\zeta,1\} 0 ≤ V H ζ ≤ min { H ζ , 1 } C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N )
We first show that the maximum and capped sum of the two components have the same pullback limit. The quantitative bound is uniform for bounded measure masses at a fixed r < 1 r<1 r < 1
Lemma 6.1. Let 0 ≤ r < 1 0 \le r < 1 0 ≤ r < 1 η , ζ \eta,\zeta η , ζ S N − 1 \mathbb{S}^{N-1} S N − 1 B ‾ r \overline{B}_r B r
u ‾ = max { u η , V H ζ } , u ‾ = min { u η + V H ζ , 1 } . \underline{u}=\max\{u_\eta,VH_\zeta\}, \qquad\overline{u}=\min\{u_\eta+VH_\zeta,1\}. u = max { u η , V H ζ } , u = min { u η + V H ζ , 1 } . For R ≥ 1 R\ge1 R ≥ 1 T ≥ 2 R + 2 T\ge2R+2 T ≥ 2 R + 2
sup ( t , x ) ∈ [ − R , R ] × B ‾ R ( 0 ) ( L T + t [ u ‾ ( − T , ⋅ ) − u ‾ ( − T , ⋅ ) ] ) ( x ) ≤ K R T ( N + 1 ) / 2 exp ( − f ′ ( 0 ) ( 1 − r ) 2 T 4 ) . (133) \sup_{(t,x)\in[-R,R]\times\overline{B}_R(0)}\left(L_{T+t}[\overline{u}(-T,\cdot)-\underline{u}(-T,\cdot)]\right)(x)\le K_R T^{(N+1)/2}\exp\left(-\frac{f'(0)(1-r)^2T}{4}\right). \tag*{(133)} ( t , x ) ∈ [ − R , R ] × B R ( 0 ) sup ( L T + t [ u ( − T , ⋅ ) − u ( − T , ⋅ )] ) ( x ) ≤ K R T ( N + 1 ) /2 exp ( − 4 f ′ ( 0 ) ( 1 − r ) 2 T ) . (133) where K R K_R K R f f f N N N r r r R R R η ( S N − 1 ) \eta(\mathbb{S}^{N-1}) η ( S N − 1 ) ζ ( B ‾ r ) \zeta(\overline{B}_r) ζ ( B r ) T → ∞ T\to\infty T → ∞
S t + T [ u ‾ ( − T , ⋅ ) ] ↗ u η ⊕ V H ζ , S t + T [ u ‾ ( − T , ⋅ ) ] ↘ u η ⊕ V H ζ , in C l o c 1 , 2 ( R × R N ) . \begin{aligned}
S_{t+T}[\underline{u}(-T,\cdot)]&\nearrow u_\eta\oplus VH_\zeta,\\
S_{t+T}[\overline{u}(-T,\cdot)]&\searrow u_\eta\oplus VH_\zeta,
\end{aligned}
\qquad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N). S t + T [ u ( − T , ⋅ )] S t + T [ u ( − T , ⋅ )] ↗ u η ⊕ V H ζ , ↘ u η ⊕ V H ζ , in C loc 1 , 2 ( R × R N ) . Furthermore,
W ∈ E , u ‾ ≤ W ≤ u ‾ ⟹ W = u η ⊕ V H ζ . W\in\mathcal{E},\qquad\underline{u}\le W\le\overline{u}\quad\Longrightarrow\quad W=u_\eta\oplus VH_\zeta. W ∈ E , u ≤ W ≤ u ⟹ W = u η ⊕ V H ζ . Proof. If either η \eta η ζ \zeta ζ u ‾ = u ‾ \underline{u}=\overline{u} u = u (131) ,
u η ( 0 , 0 ) ≤ U r a d ( ln max { η ( S N − 1 ) , 1 } 2 f ′ ( 0 ) , 0 ) = : θ < 1. u_\eta(0,0)\le U^{\mathrm{rad}}\left(\frac{\ln\max\{\eta(\mathbb{S}^{N-1}),1\}}{2f'(0)},0\right)=:\theta<1. u η ( 0 , 0 ) ≤ U rad ( 2 f ′ ( 0 ) ln max { η ( S N − 1 ) , 1 } , 0 ) =: θ < 1. By (43) and (132) , one has
u η ( − T , 0 ) = D e − 2 f ′ ( 0 ) T u η ( 0 , 0 ) ≤ ϕ ∗ ( − c ∗ T + ϕ ∗ − 1 ( θ ) ) ≤ K ( 1 + T ) e − 2 f ′ ( 0 ) T for T ≥ 1. u_\eta(-T,0)=D_{e^{-2f'(0)T}}u_\eta(0,0)\le\phi_*\left(-c_*T+\phi_*^{-1}(\theta)\right)\le K(1+T)e^{-2f'(0)T}\qquad\text{for }T\ge1. u η ( − T , 0 ) = D e − 2 f ′ ( 0 ) T u η ( 0 , 0 ) ≤ ϕ ∗ ( − c ∗ T + ϕ ∗ − 1 ( θ ) ) ≤ K ( 1 + T ) e − 2 f ′ ( 0 ) T for T ≥ 1. Here K K K θ \theta θ ϕ ∗ \phi_* ϕ ∗ ∣ ∇ ln u η ∣ ≤ λ ∗ |\nabla\ln u_\eta|\le\lambda_* ∣∇ ln u η ∣ ≤ λ ∗
u η ( − T , y ) ≤ K ( 1 + T ) e − 2 f ′ ( 0 ) T + λ ∗ ∣ y ∣ for y ∈ R N , T ≥ 1. (134) u_\eta(-T,y)\le K(1+T)e^{-2f'(0)T+\lambda_*|y|}\qquad\text{for }y\in\mathbb{R}^N,\ T\ge1. \tag*{(134)} u η ( − T , y ) ≤ K ( 1 + T ) e − 2 f ′ ( 0 ) T + λ ∗ ∣ y ∣ for y ∈ R N , T ≥ 1. (134) By V H ζ ≤ H ζ VH_\zeta\le H_\zeta V H ζ ≤ H ζ H ζ H_\zeta H ζ
V H ζ ( − T , y ) ≤ H ζ ( − T , y ) = ∫ B ‾ r e λ ∗ y ⋅ ξ − f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) T d ζ ( ξ ) ≤ ζ ( B ‾ r ) sup 0 ≤ a ≤ r exp { λ ∗ a ∣ y ∣ − f ′ ( 0 ) ( 1 + a 2 ) T } . (135) \begin{aligned}
VH_\zeta(-T,y)&\le H_\zeta(-T,y)=\int_{\overline{B}_r}e^{\lambda_*y\cdot\xi-f'(0)(1+|\xi|^2)T}\,d\zeta(\xi)\\
&\le\zeta(\overline{B}_r)\sup_{0\le a\le r}\exp\{\lambda_*a|y|-f'(0)(1+a^2)T\}. \tag*{(135)}
\end{aligned} V H ζ ( − T , y ) ≤ H ζ ( − T , y ) = ∫ B r e λ ∗ y ⋅ ξ − f ′ ( 0 ) ( 1 + ∣ ξ ∣ 2 ) T d ζ ( ξ ) ≤ ζ ( B r ) 0 ≤ a ≤ r sup exp { λ ∗ a ∣ y ∣ − f ′ ( 0 ) ( 1 + a 2 ) T } . (135) Since 0 ≤ u η , V H ζ ≤ 1 0\le u_\eta,VH_\zeta\le1 0 ≤ u η , V H ζ ≤ 1
0 ≤ u ‾ − u ‾ ≤ min { u η , V H ζ } ≤ u η V H ζ . 0\le\overline{u}-\underline{u}\le\min\{u_\eta,VH_\zeta\}\le\sqrt{u_\eta VH_\zeta}. 0 ≤ u − u ≤ min { u η , V H ζ } ≤ u η V H ζ . For ∣ t ∣ , ∣ x ∣ ≤ R |t|,|x|\le R ∣ t ∣ , ∣ x ∣ ≤ R T ≥ 2 R + 2 T\ge2R+2 T ≥ 2 R + 2 ρ = ∣ y − x ∣ \rho=|y-x| ρ = ∣ y − x ∣ ∣ y ∣ ≤ R + ρ |y|\le R+\rho ∣ y ∣ ≤ R + ρ (134) –(135) give
( L T + t [ ( u ‾ − u ‾ ) ( − T , ⋅ ) ] ) ( x ) ≤ ( L T + t [ u η ( − T , ⋅ ) V H ζ ( − T , ⋅ ) ] ) ( x ) ≤ K R ( 1 + T ) 1 / 2 ( T + t ) − N / 2 ∫ 0 ∞ ρ N − 1 sup 0 ≤ a ≤ r e E a ( ρ ) d ρ , \begin{aligned}
\left(L_{T+t}[(\overline{u}-\underline{u})(-T,\cdot)]\right)(x)
&\le\left(L_{T+t}\left[\sqrt{u_\eta(-T,\cdot)VH_\zeta(-T,\cdot)}\right]\right)(x)\\
&\le K_R(1+T)^{1/2}(T+t)^{-N/2}\int_0^\infty\rho^{N-1}\sup_{0\le a\le r}e^{E_a(\rho)}\,d\rho,
\end{aligned} ( L T + t [( u − u ) ( − T , ⋅ )] ) ( x ) ≤ ( L T + t [ u η ( − T , ⋅ ) V H ζ ( − T , ⋅ ) ] ) ( x ) ≤ K R ( 1 + T ) 1/2 ( T + t ) − N /2 ∫ 0 ∞ ρ N − 1 0 ≤ a ≤ r sup e E a ( ρ ) d ρ , where, with I T + t = [ λ ∗ ( T + t ) , λ ∗ ( 1 + r ) ( T + t ) ] I_{T+t}=[\lambda_*(T+t),\lambda_*(1+r)(T+t)] I T + t = [ λ ∗ ( T + t ) , λ ∗ ( 1 + r ) ( T + t )]
E a ( ρ ) = − ρ 2 4 ( T + t ) + λ ∗ ( 1 + a ) ρ 2 + f ′ ( 0 ) ( 3 + a 2 ) T 2 + f ′ ( 0 ) ( T + t ) = − ( ρ − λ ∗ ( 1 + a ) ( T + t ) ) 2 4 ( T + t ) − f ′ ( 0 ) ( 1 − a ) 2 T 4 + f ′ ( 0 ) t ( 1 + ( 1 + a ) 2 4 ) ≤ − dist ( ρ , I T + t ) 2 4 ( T + t ) − f ′ ( 0 ) ( 1 − r ) 2 T 4 + 2 f ′ ( 0 ) R . \begin{aligned}
E_a(\rho)&=-\frac{\rho^2}{4(T+t)}+\frac{\lambda_*(1+a)\rho}{2}+\frac{f'(0)(3+a^2)T}{2}+f'(0)(T+t)\\
&=-\frac{(\rho-\lambda_*(1+a)(T+t))^2}{4(T+t)}-\frac{f'(0)(1-a)^2T}{4}+f'(0)t\left(1+\frac{(1+a)^2}{4}\right)\\
&\leq-\frac{\operatorname{dist}(\rho,I_{T+t})^2}{4(T+t)}-\frac{f'(0)(1-r)^2T}{4}+2f'(0)R.
\end{aligned} E a ( ρ ) = − 4 ( T + t ) ρ 2 + 2 λ ∗ ( 1 + a ) ρ + 2 f ′ ( 0 ) ( 3 + a 2 ) T + f ′ ( 0 ) ( T + t ) = − 4 ( T + t ) ( ρ − λ ∗ ( 1 + a ) ( T + t ) ) 2 − 4 f ′ ( 0 ) ( 1 − a ) 2 T + f ′ ( 0 ) t ( 1 + 4 ( 1 + a ) 2 ) ≤ − 4 ( T + t ) dist ( ρ , I T + t ) 2 − 4 f ′ ( 0 ) ( 1 − r ) 2 T + 2 f ′ ( 0 ) R . Since dist ( ρ , I T + t ) ≥ ρ / 2 \operatorname{dist}(\rho,I_{T+t})\geq\rho/2 dist ( ρ , I T + t ) ≥ ρ /2 ρ ≥ 4 λ ∗ ( T + t ) \rho\geq4\lambda_*(T+t) ρ ≥ 4 λ ∗ ( T + t )
∫ 0 ∞ ρ N − 1 exp ( − dist ( ρ , I T + t ) 2 4 ( T + t ) ) d ρ ≤ ( 4 λ ∗ ( T + t ) ) N N + ∫ 0 ∞ ρ N − 1 e − ρ 2 / ( 16 ( T + t ) ) d ρ ≤ K N , f ( T + t ) N . (136) \begin{aligned}
\int_0^\infty\rho^{N-1}\exp\left(-\frac{\operatorname{dist}(\rho,I_{T+t})^2}{4(T+t)}\right)\,d\rho
&\leq\frac{(4\lambda_*(T+t))^N}{N}+\int_0^\infty\rho^{N-1}e^{-\rho^2/(16(T+t))}\,d\rho\\
&\leq K_{N,f}(T+t)^N. \tag*{(136)}
\end{aligned} ∫ 0 ∞ ρ N − 1 exp ( − 4 ( T + t ) dist ( ρ , I T + t ) 2 ) d ρ ≤ N ( 4 λ ∗ ( T + t ) ) N + ∫ 0 ∞ ρ N − 1 e − ρ 2 / ( 16 ( T + t )) d ρ ≤ K N , f ( T + t ) N . (136) Together with T / 2 ≤ T + t ≤ 3 T / 2 T/2\leq T+t\leq3T/2 T /2 ≤ T + t ≤ 3 T /2 (133) .
The functions u ‾ , u ‾ \underline{u},\overline{u} u , u
f ( u η ) + f ( V H ζ ) − f ( u η + V H ζ ) ≥ 0 f(u_\eta)+f(V_{H_\zeta})-f(u_\eta+V_{H_\zeta})\geq0 f ( u η ) + f ( V H ζ ) − f ( u η + V H ζ ) ≥ 0 by concavity of f f f f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0
The comparison principle gives
0 ≤ S T + t [ u ‾ ( − T , ⋅ ) ] − S T + t [ u ‾ ( − T , ⋅ ) ] ≤ L T + t [ ( u ‾ − u ‾ ) ( − T , ⋅ ) ] → 0 \begin{aligned}
0&\leq S_{T+t}[\overline{u}(-T,\cdot)]-S_{T+t}[\underline{u}(-T,\cdot)]\\
&\leq L_{T+t}[(\overline{u}-\underline{u})(-T,\cdot)]\to0
\end{aligned} 0 ≤ S T + t [ u ( − T , ⋅ )] − S T + t [ u ( − T , ⋅ )] ≤ L T + t [( u − u ) ( − T , ⋅ )] → 0 locally uniformly as T → ∞ T\to\infty T → ∞ u ‾ ≤ u η ⊕ V H ζ ≤ u ‾ \underline{u}\leq u_\eta\oplus V_{H_\zeta}\leq\overline{u} u ≤ u η ⊕ V H ζ ≤ u
S T + t [ u ‾ ( − T , ⋅ ) ] ↗ u η ⊕ V H ζ , S T + t [ u ‾ ( − T , ⋅ ) ] ↘ u η ⊕ V H ζ S_{T+t}[\underline{u}(-T,\cdot)]\nearrow u_\eta\oplus V_{H_\zeta},
\qquad
S_{T+t}[\overline{u}(-T,\cdot)]\searrow u_\eta\oplus V_{H_\zeta} S T + t [ u ( − T , ⋅ )] ↗ u η ⊕ V H ζ , S T + t [ u ( − T , ⋅ )] ↘ u η ⊕ V H ζ in C l o c 1 , 2 ( R × R N ) C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N) C loc 1 , 2 ( R × R N ) T → ∞ T\to\infty T → ∞
For W ∈ E W\in\mathcal{E} W ∈ E u ‾ ≤ W ≤ u ‾ \underline{u}\leq W\leq\overline{u} u ≤ W ≤ u
S T + t [ u ‾ ( − T , ⋅ ) ] ≤ W ( t , ⋅ ) ≤ S T + t [ u ‾ ( − T , ⋅ ) ] . S_{T+t}[\underline{u}(-T,\cdot)]\leq W(t,\cdot)\leq S_{T+t}[\overline{u}(-T,\cdot)]. S T + t [ u ( − T , ⋅ )] ≤ W ( t , ⋅ ) ≤ S T + t [ u ( − T , ⋅ )] . Letting T → ∞ T\to\infty T → ∞ W = u η ⊕ V H ζ W=u_\eta\oplus V_{H_\zeta} W = u η ⊕ V H ζ
At a fixed strict gradient gap, the preceding comparison estimates allow weak approximation of the critical and subcritical measures simultaneously.
Proposition 6.2. Fix 0 ≤ r < 1 0\leq r<1 0 ≤ r < 1 η n ⇀ η \eta_n\rightharpoonup\eta η n ⇀ η S N − 1 \mathbb{S}^{N-1} S N − 1 ζ n ⇀ ζ \zeta_n\rightharpoonup\zeta ζ n ⇀ ζ B ‾ r \overline{B}_r B r
u η n ⊕ V H ζ n → u η ⊕ V H ζ in C l o c 1 , 2 ( R × R N ) . u_{\eta_n}\oplus V_{H_{\zeta_n}}\to u_\eta\oplus V_{H_\zeta}\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N). u η n ⊕ V H ζ n → u η ⊕ V H ζ in C loc 1 , 2 ( R × R N ) . Proof. By Proposition 5.9 and Theorem 3.5 ,
u η n → u η , V H ζ n → V H ζ in C l o c 1 , 2 ( R × R N ) . u_{\eta_n}\to u_\eta,\qquad V_{H_{\zeta_n}}\to V_{H_\zeta}\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N). u η n → u η , V H ζ n → V H ζ in C loc 1 , 2 ( R × R N ) . Since 0 ≤ u η n ⊕ V H ζ n ≤ 1 0\leq u_{\eta_n}\oplus V_{H_{\zeta_n}}\leq1 0 ≤ u η n ⊕ V H ζ n ≤ 1
u η n j ⊕ V H ζ n j → W in C l o c 1 , 2 ( R × R N ) , W ∈ E . u_{\eta_{n_j}}\oplus V_{H_{\zeta_{n_j}}}\to W\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^N),\qquad W\in\mathcal{E}. u η n j ⊕ V H ζ n j → W in C loc 1 , 2 ( R × R N ) , W ∈ E . Passing to the limit in
max { u η n , V H ζ n } ≤ u η n ⊕ V H ζ n ≤ min { u η n + V H ζ n , 1 } \max\{u_{\eta_n},V_{H\zeta_n}\}\leq u_{\eta_n}\oplus V_{H\zeta_n}\leq\min\{u_{\eta_n}+V_{H\zeta_n},1\} max { u η n , V H ζ n } ≤ u η n ⊕ V H ζ n ≤ min { u η n + V H ζ n , 1 } yields
max { u η , V H ζ } ≤ W ≤ min { u η + V H ζ , 1 } . \max\{u_{\eta},V_{H\zeta}\}\leq W\leq\min\{u_{\eta}+V_{H\zeta},1\}. max { u η , V H ζ } ≤ W ≤ min { u η + V H ζ , 1 } . Lemma 6.1 gives W = u η ⊕ V H ζ W=u_{\eta}\oplus V_{H\zeta} W = u η ⊕ V H ζ
Proof of the closure theorem
We now apply the preceding results to an arbitrary entire solution. After fixing one approximation of its noncritical component, we approximate both measures by finite atomic measures and then choose a diagonal sequence to recover the full solution.
Proof of Theorem 1.1 . For U ≡ 0 U\equiv0 U ≡ 0 U ≡ 1 U\equiv1 U ≡ 1
ξ f ( t − j ) → 0 , ξ f ( t + j ) → 1 in C l o c 1 , 2 ( R × R N ) . \xi_f(t-j)\to0,\qquad\xi_f(t+j)\to1\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^{N}). ξ f ( t − j ) → 0 , ξ f ( t + j ) → 1 in C loc 1 , 2 ( R × R N ) . By the strong maximum principle, it remains to consider 0 < U < 1 0<U<1 0 < U < 1
Proposition 4.3 gives U ∗ , Z , A j ∈ C U_*,Z,A_j\in C U ∗ , Z , A j ∈ C
U = Z ⊕ U ∗ , A j ↗ Z in C l o c 1 , 2 ( R × R N ) , U=Z\oplus U_*,\qquad A_j\nearrow Z\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^{N}), U = Z ⊕ U ∗ , A j ↗ Z in C loc 1 , 2 ( R × R N ) , ∣ ∇ ln A j ∣ ≤ λ ∗ r j , 0 ≤ r j < 1 if A j ≠ 0 , |\nabla\ln A_j|\leq\lambda_*r_j,\qquad0\leq r_j<1\quad\text{if }A_j\neq0, ∣∇ ln A j ∣ ≤ λ ∗ r j , 0 ≤ r j < 1 if A j = 0 , D a U ∗ ( t , x ) = U ∗ ( t + ln a 2 f ′ ( 0 ) , x ) for a > 0. D_aU_*(t,x)=U_*\left(t+\frac{\ln a}{2f'(0)},x\right)\quad\text{for }a>0. D a U ∗ ( t , x ) = U ∗ ( t + 2 f ′ ( 0 ) ln a , x ) for a > 0. By Theorems 3.8 , 3.5 , and 5.8 , there are finite nonnegative Borel measures ζ j , η \zeta_j,\eta ζ j , η
A j = V H ζ j , supp ζ j ⊂ B ‾ r j , U ∗ = u η , supp η ⊂ S N − 1 . A_j=V_{H\zeta_j},\qquad\operatorname{supp}\zeta_j\subset\overline{B}_{r_j},\qquad U_*=u_\eta,\qquad\operatorname{supp}\eta\subset S^{N-1}. A j = V H ζ j , supp ζ j ⊂ B r j , U ∗ = u η , supp η ⊂ S N − 1 . For A j = 0 A_j=0 A j = 0 ζ j = 0 \zeta_j=0 ζ j = 0 r j = 0 r_j=0 r j = 0 U ∗ = 0 U_*=0 U ∗ = 0 η = 0 \eta=0 η = 0 Corollary 2.17 yields
U j : = A j ⊕ U ∗ ↗ U in C l o c 1 , 2 ( R × R N ) . (137) U_j:=A_j\oplus U_*\nearrow U\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^{N}). \tag*{(137)} U j := A j ⊕ U ∗ ↗ U in C loc 1 , 2 ( R × R N ) . (137) Since U ( 0 , 0 ) > 0 U(0,0)>0 U ( 0 , 0 ) > 0 U j ≠ 0 U_j\neq0 U j = 0
For each fixed j j j
ζ j , n ⇀ ζ j , ζ j , n ( B ‾ r j ) = ζ j ( B ‾ r j ) , \zeta_{j,n}\rightharpoonup\zeta_j,\qquad\zeta_{j,n}(\overline{B}_{r_j})=\zeta_j(\overline{B}_{r_j}), ζ j , n ⇀ ζ j , ζ j , n ( B r j ) = ζ j ( B r j ) , η n ⇀ η , η n ( S N − 1 ) = η ( S N − 1 ) , \eta_n\rightharpoonup\eta,\qquad\eta_n(S^{N-1})=\eta(S^{N-1}), η n ⇀ η , η n ( S N − 1 ) = η ( S N − 1 ) , supp ζ j , n ⊂ B ‾ r j , supp η n ⊂ S N − 1 . \operatorname{supp}\zeta_{j,n}\subset\overline{B}_{r_j},\qquad\operatorname{supp}\eta_n\subset S^{N-1}. supp ζ j , n ⊂ B r j , supp η n ⊂ S N − 1 . These are obtained by placing each cell’s mass at one point of a finite Borel partition of diameter at most 1 / n 1/n 1/ n Proposition 6.2 gives
F j , n : = V H ζ j , n ⊕ u η n → U j in C l o c 1 , 2 ( R × R N ) as n → ∞ . F_{j,n}:=V_{H\zeta_{j,n}}\oplus u_{\eta_n}\to U_j\quad\text{in }C_{\mathrm{loc}}^{1,2}(\mathbb{R}\times\mathbb{R}^{N})\quad\text{as }n\to\infty. F j , n := V H ζ j , n ⊕ u η n → U j in C loc 1 , 2 ( R × R N ) as n → ∞. We now identify each F j , n F_{j,n} F j , n ( j , n ) (j,n) ( j , n ) ζ = ζ j , n \zeta=\zeta_{j,n} ζ = ζ j , n η = η n \eta=\eta_n η = η n
ζ = a 0 δ 0 + ∑ i = 1 p a i δ ξ i , η = ∑ ℓ = 1 k b ℓ δ ν ℓ , \zeta=a_0\delta_0+\sum_{i=1}^{p}a_i\delta_{\xi_i},\qquad\eta=\sum_{\ell=1}^{k}b_\ell\delta_{\nu_\ell}, ζ = a 0 δ 0 + i = 1 ∑ p a i δ ξ i , η = ℓ = 1 ∑ k b ℓ δ ν ℓ , where a 0 ≥ 0 a_0 \ge0 a 0 ≥ 0 a i , b j > 0 a_i,b_j>0 a i , b j > 0 0 < ∣ ξ i ∣ < 1 0<|\xi_i|<1 0 < ∣ ξ i ∣ < 1 ∣ ν j ∣ = 1 |\nu_j|=1 ∣ ν j ∣ = 1 a 0 + ∑ i a i + ∑ j b j > 0 a_0+\sum_i a_i+\sum_j b_j>0 a 0 + ∑ i a i + ∑ j b j > 0
ρ i : = ∣ ξ i ∣ , e i = ξ i ∣ ξ i ∣ , c i = λ ∗ ( ∣ ξ i ∣ + ∣ ξ i ∣ − 1 ) > c ∗ , h i = ln a i λ ∗ ∣ ξ i ∣ . \rho_i:=|\xi_i|,\qquad e_i=\frac{\xi_i}{|\xi_i|},\qquad c_i=\lambda_*(|\xi_i|+|\xi_i|^{-1})>c_*,\qquad h_i=\frac{\ln a_i}{\lambda_*|\xi_i|}. ρ i := ∣ ξ i ∣ , e i = ∣ ξ i ∣ ξ i , c i = λ ∗ ( ∣ ξ i ∣ + ∣ ξ i ∣ − 1 ) > c ∗ , h i = λ ∗ ∣ ξ i ∣ ln a i . Use the normalized homogeneous orbit ξ f \xi_f ξ f (8) . The corresponding profiles are
v i ( t , x ) = ϕ c i ( x ⋅ e i + c i t + h i ) , v 0 ( t ) = ξ f ( t + ln a 0 f ′ ( 0 ) ) if a 0 > 0 , v_i(t,x)=\phi_{c_i}(x\cdot e_i+c_it+h_i),\qquad v_0(t)=\xi_f\left(t+\frac{\ln a_0}{f'(0)}\right)\quad\text{if }a_0>0, v i ( t , x ) = ϕ c i ( x ⋅ e i + c i t + h i ) , v 0 ( t ) = ξ f ( t + f ′ ( 0 ) ln a 0 ) if a 0 > 0 , and
w j ( t , x ) = ϕ ∗ ( x ⋅ ν j + c ∗ t + λ ∗ − 1 ln b j ) . (138) w_j(t,x)=\phi_*\left(x\cdot\nu_j+c_*t+\lambda_*^{-1}\ln b_j\right). \tag*{(138)} w j ( t , x ) = ϕ ∗ ( x ⋅ ν j + c ∗ t + λ ∗ − 1 ln b j ) . (138) Here
ϕ c i ( s ) ∼ e λ ∗ ∣ ξ i ∣ s , ξ f ( s ) ∼ e f ′ ( 0 ) s as s → − ∞ . \phi_{c_i}(s)\sim e^{\lambda_*|\xi_i|s},\qquad\xi_f(s)\sim e^{f'(0)s}\qquad\text{as }s\to-\infty. ϕ c i ( s ) ∼ e λ ∗ ∣ ξ i ∣ s , ξ f ( s ) ∼ e f ′ ( 0 ) s as s → − ∞. In all maxima and sums below, v 0 v_0 v 0 a 0 > 0 a_0>0 a 0 > 0 Theorem 3.5 ,
max i v i ≤ V H ζ ≤ min { ∑ i v i , 1 } . \max_i v_i\le V_{H_\zeta}\le\min\left\{\sum_i v_i,1\right\}. i max v i ≤ V H ζ ≤ min { i ∑ v i , 1 } . For k ≥ 1 k\ge1 k ≥ 1 g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) g(q)=\phi_*(\lambda_*^{-1}\ln q) g ( q ) = ϕ ∗ ( λ ∗ − 1 ln q ) g ( 0 ) = 0 g(0)=0 g ( 0 ) = 0
max j w j ≤ g ( H η ) ≤ ∑ j w j . \max_j w_j\le g(H_\eta)\le\sum_j w_j. j max w j ≤ g ( H η ) ≤ j ∑ w j . Comparison in (114) therefore yields
max j w j ≤ u η ≤ min { ∑ j w j , 1 } . \max_j w_j\le u_\eta\le\min\left\{\sum_j w_j,1\right\}. j max w j ≤ u η ≤ min { j ∑ w j , 1 } . The definition of ⊕ \oplus ⊕
max { max i v i , max j w j } ≤ V H ζ ⊕ u η ≤ min { ∑ i v i + ∑ j w j , 1 } . \max\{\max_i v_i,\max_j w_j\}\le V_{H_\zeta}\oplus u_\eta\le\min\left\{\sum_i v_i+\sum_j w_j,1\right\}. max { i max v i , j max w j } ≤ V H ζ ⊕ u η ≤ min { i ∑ v i + j ∑ w j , 1 } . By Theorem 3.4 , V H ζ ⊕ u η V_{H_\zeta}\oplus u_\eta V H ζ ⊕ u η v i v_i v i w j w_j w j v 0 v_0 v 0 a 0 > 0 a_0>0 a 0 > 0
( c i , ln a i λ ∗ ∣ ξ i ∣ ) , ( c ∗ , λ ∗ − 1 ln b j ) , \left(c_i,\frac{\ln a_i}{\lambda_*|\xi_i|}\right),\qquad\left(c_*,\lambda_*^{-1}\ln b_j\right), ( c i , λ ∗ ∣ ξ i ∣ ln a i ) , ( c ∗ , λ ∗ − 1 ln b j ) , with homogeneous time shift ln a 0 / f ′ ( 0 ) \ln a_0/f'(0) ln a 0 / f ′ ( 0 ) a 0 > 0 a_0>0 a 0 > 0
Here is also the precise original measure corresponding to this solution. Our normalized critical profile ϕ ∗ \phi_* ϕ ∗ [28 ] . Thus the original critical phases are λ ∗ − 1 ln b j \lambda_*^{-1}\ln b_j λ ∗ − 1 ln b j c ∗ ln m j c_*\ln m_j c ∗ ln m j m j m_j m j u α m ( t , x ) = u m ( t + ln α , x ) u_{\alpha m}(t,x)=u_m(t+\ln\alpha,x) u α m ( t , x ) = u m ( t + ln α , x ) [28 ] , Theorem 1.2(i),(iii) and Section 3.1. Consequently the original critical mass is m j = b j 1 / ( 2 f ′ ( 0 ) ) m_j=b_j^{1/(2f'(0))} m j = b j 1/ ( 2 f ′ ( 0 )) 1 / λ ∗ 1/\lambda_* 1/ λ ∗
If a 0 + ∑ i a i > 0 a_0+\sum_i a_i>0 a 0 + ∑ i a i > 0 M > 0 M>0 M > 0
a 0 M f ′ ( 0 ) + ∑ i = 1 p a i M f ′ ( 0 ) ( 1 + ρ i 2 ) = 1. (139) \frac{a_0}{M^{f'(0)}}+\sum_{i=1}^{p}\frac{a_i}{M^{f'(0)}(1+\rho_i^2)}=1. \tag*{(139)} M f ′ ( 0 ) a 0 + i = 1 ∑ p M f ′ ( 0 ) ( 1 + ρ i 2 ) a i = 1. (139) The left side is continuous and strictly decreasing from + ∞ +\infty + ∞ 0 0 0 M M M
m = ∑ ℓ = 1 k b j 1 / ( 2 f ′ ( 0 ) ) δ ( ν j , c ∗ ) + ∑ i = 1 p a i M 1 − f ′ ( 0 ) ( 1 + ρ i 2 ) δ ( e i , c i ) + a 0 M 1 − f ′ ( 0 ) δ ∞ . (140) \mathbf{m}=\sum_{\ell=1}^{k}b_j^{1/(2f'(0))}\delta_{(\nu_j,c_*)}+\sum_{i=1}^{p}a_iM^{1-f'(0)}(1+\rho_i^2)\delta_{(e_i,c_i)}+a_0M^{1-f'(0)}\delta_{\infty}. \tag*{(140)} m = ℓ = 1 ∑ k b j 1/ ( 2 f ′ ( 0 )) δ ( ν j , c ∗ ) + i = 1 ∑ p a i M 1 − f ′ ( 0 ) ( 1 + ρ i 2 ) δ ( e i , c i ) + a 0 M 1 − f ′ ( 0 ) δ ∞ . (140) When the noncritical list is empty, omit M M M (139) ,
m ( X ∖ ( S N − 1 × { c ∗ } ) ) = ∑ i = 1 p a i M 1 − f ′ ( 0 ) ( 1 + ρ i 2 ) + a 0 M 1 − f ′ ( 0 ) = M , \mathop{\mathrm{m}}\left(X\setminus\left(S^{N-1}\times\{c_*\}\right)\right)=\sum_{i=1}^{p}a_iM^{1-f'(0)}(1+\rho_i^2)+a_0M^{1-f'(0)}=M, m ( X ∖ ( S N − 1 × { c ∗ } ) ) = i = 1 ∑ p a i M 1 − f ′ ( 0 ) ( 1 + ρ i 2 ) + a 0 M 1 − f ′ ( 0 ) = M , m ( { ( e i , c i ) } ) M λ ( c i ) c i − 1 = a i , m ( { ∞ } ) M f ′ ( 0 ) − 1 = a 0 . \mathop{\mathrm{m}}\left(\{(e_i,c_i)\}\right)M^{\lambda(c_i)c_i-1}=a_i,\qquad\mathop{\mathrm{m}}(\{\infty\})M^{f'(0)-1}=a_0. m ( {( e i , c i )} ) M λ ( c i ) c i − 1 = a i , m ({ ∞ }) M f ′ ( 0 ) − 1 = a 0 . Here λ ( c i ) c i = f ′ ( 0 ) ( 1 + ρ i 2 ) \lambda(c_i)c_i=f'(0)(1+\rho_i^2) λ ( c i ) c i = f ′ ( 0 ) ( 1 + ρ i 2 ) [28 Section 3.9, Lemma 3.7] and uniqueness in Theorem 3.4 give
V H ζ ⊕ u η = u m , m ∈ M . V_{H_\zeta}\oplus u_\eta=u_{\mathbf{m}},\qquad\mathbf{m}\in\mathcal{M}. V H ζ ⊕ u η = u m , m ∈ M . The masses entering Lemma 6.1 are
η ( S N − 1 ) = ∑ j b j , ζ ( B 1 ‾ ) = a 0 + ∑ i a i , \eta(S^{N-1})=\sum_j b_j,\qquad\zeta(\overline{B_1})=a_0+\sum_i a_i, η ( S N − 1 ) = j ∑ b j , ζ ( B 1 ) = a 0 + i ∑ a i , not the original critical mass ∑ j b j 1 / ( 2 f ′ ( 0 ) ) \sum_j b_j^{1/(2f'(0))} ∑ j b j 1/ ( 2 f ′ ( 0 )) m \mathbf{m} m M \mathcal{M} M m ( X ) \mathbf{m}(X) m ( X ) F j , n F_{j,n} F j , n n ( j ) n(j) n ( j )
∥ F j , n ( j ) − U j ∥ C 1 , 2 ( [ − j , j ] × B j ( 0 ) ‾ ) < 1 / j , F j : = F j , n ( j ) . \|F_{j,n(j)}-U_j\|_{C^{1,2}([-j,j]\times\overline{B_j(0)})}<1/j,\qquad F_j:=F_{j,n(j)}. ∥ F j , n ( j ) − U j ∥ C 1 , 2 ([ − j , j ] × B j ( 0 ) ) < 1/ j , F j := F j , n ( j ) . For every fixed R ≥ 1 R\ge1 R ≥ 1 j ≥ R j\ge R j ≥ R
∥ F j − U ∥ C 1 , 2 ( [ − R , R ] × B R ( 0 ) ‾ ) ≤ ∥ F j − U j ∥ C 1 , 2 ( [ − j , j ] × B j ( 0 ) ‾ ) + ∥ U j − U ∥ C 1 , 2 ( [ − R , R ] × B R ( 0 ) ‾ ) < 1 / j + ∥ U j − U ∥ C 1 , 2 ( [ − R , R ] × B R ( 0 ) ‾ ) → 0. \begin{aligned}
\|F_j-U\|_{C^{1,2}([-R,R]\times\overline{B_R(0)})}
&\le\|F_j-U_j\|_{C^{1,2}([-j,j]\times\overline{B_j(0)})}\\
&\quad+\|U_j-U\|_{C^{1,2}([-R,R]\times\overline{B_R(0)})}\\
&<1/j+\|U_j-U\|_{C^{1,2}([-R,R]\times\overline{B_R(0)})}\to0.
\end{aligned} ∥ F j − U ∥ C 1 , 2 ([ − R , R ] × B R ( 0 ) ) ≤ ∥ F j − U j ∥ C 1 , 2 ([ − j , j ] × B j ( 0 ) ) + ∥ U j − U ∥ C 1 , 2 ([ − R , R ] × B R ( 0 ) ) < 1/ j + ∥ U j − U ∥ C 1 , 2 ([ − R , R ] × B R ( 0 ) ) → 0. By (140) , F j = u ϑ j F_j=u_{\vartheta_j} F j = u ϑ j ϑ j \vartheta_j ϑ j X = S N − 1 × [ c ∗ , ∞ ) ∪ { ∞ } X=S^{N-1}\times[c_*,\infty)\cup\{\infty\} X = S N − 1 × [ c ∗ , ∞ ) ∪ { ∞ } u ϑ j → U u_{\vartheta_j}\to U u ϑ j → U C l o c 1 , 2 C_{\mathrm{loc}}^{1,2} C loc 1 , 2 [28 ] . The reverse inclusion follows from parabolic regularity and passage to the limit in (1) .
