Introduction and main results
In this paper, we consider the semilinear Hardy-Hénon parabolic equation
u t = Δ u + ∣ x ∣ l ∣ u ∣ α u , (1) u_t=\Delta u+|x|^{l}|u|^{\alpha}u, \tag*{(1)} u t = Δ u + ∣ x ∣ l ∣ u ∣ α u , (1) with initial data
u ( 0 ) = u 0 ≥ 0 , (2) u(0)=u_0\ge0, \tag*{(2)} u ( 0 ) = u 0 ≥ 0 , (2) where u = u ( t , x ) u=u(t,x) u = u ( t , x ) ( t , x ) ∈ ( 0 , T ) × D (t,x)\in(0,T)\times D ( t , x ) ∈ ( 0 , T ) × D T > 0 T>0 T > 0 D = R N D=\mathbb{R}^{N} D = R N l ≥ 0 l\ge0 l ≥ 0 D = R N ∖ { 0 } D=\mathbb{R}^{N}\setminus\{0\} D = R N ∖ { 0 } l < 0 l<0 l < 0 N ∈ N N\in\mathbb{N} N ∈ N l > − min { 2 , N } l>-\min\{2,N\} l > − min { 2 , N } α > 0 \alpha>0 α > 0 l < 0 l<0 l < 0 l > 0 l>0 l > 0 l = 0 l=0 l = 0 [3 , 4 , 6 , 8 , 9 , 10 , 11 , 12 , 32 , 34 , 37 ] and the references therein. We denote by T max ( u 0 ) T_{\max}(u_0) T m a x ( u 0 ) 0 < T max ( u 0 ) < ∞ 0<T_{\max}(u_0)<\infty 0 < T m a x ( u 0 ) < ∞ T max ( u 0 ) = ∞ T_{\max}(u_0)=\infty T m a x ( u 0 ) = ∞ T max ( u 0 ) = 0 T_{\max}(u_0)=0 T m a x ( u 0 ) = 0 u 0 = λ φ u_0=\lambda\varphi u 0 = λ φ φ \varphi φ λ > 0 \lambda>0 λ > 0
For the case l = 0 l=0 l = 0 [17 , 24 , 26 , 34 , 35 ] . In the case l ≠ 0 l\ne0 l = 0 [34 ] . See also [6 , 34 ] for upper bound estimates of the life-span for certain classes of initial data.
In this paper, we investigate finite-time blow-up solutions for the case l ≠ 0 l\ne0 l = 0 u 0 = λ φ u_0=\lambda\varphi u 0 = λ φ λ \lambda λ φ \varphi φ
As is standard practice, we study (1.1)–(1.2) via the associated integral equation
u ( t ) = e t Δ u 0 + ∫ 0 t e ( t − σ ) Δ [ ∣ ⋅ ∣ l ∣ u ( σ ) ∣ α u ( σ ) ] d σ (3) u(t)=e^{t\Delta}u_0+\int_0^t e^{(t-\sigma)\Delta}\left[|\cdot|^l|u(\sigma)|^\alpha u(\sigma)\right]\,d\sigma \tag*{(3)} u ( t ) = e t Δ u 0 + ∫ 0 t e ( t − σ ) Δ [ ∣ ⋅ ∣ l ∣ u ( σ ) ∣ α u ( σ ) ] d σ (3) where, for all t > 0 t>0 t > 0 e t Δ e^{t\Delta} e t Δ
e t Δ f = G ( t , ⋅ ) ∗ f , (4) e^{t\Delta}f=G(t,\cdot)*f, \tag*{(4)} e t Δ f = G ( t , ⋅ ) ∗ f , (4) with ∗ * ∗
G ( t , x ) = ( 4 π t ) − N 2 e − ∣ x ∣ 2 4 t , t > 0 , x ∈ R N . (5) G(t,x)=(4\pi t)^{-\frac{N}{2}}e^{-\frac{|x|^2}{4t}},\ t>0,\ x\in\mathbb{R}^N. \tag*{(5)} G ( t , x ) = ( 4 π t ) − 2 N e − 4 t ∣ x ∣ 2 , t > 0 , x ∈ R N . (5) A solution u u u (3) is a mild solution of the problem (1.1)–(1.2).
In what follows, for 1 ≤ q ≤ ∞ 1\le q\le\infty 1 ≤ q ≤ ∞ L q ( R N ) L^q(\mathbb{R}^N) L q ( R N ) ∥ ⋅ ∥ L q \|\cdot\|_{L^q} ∥ ⋅ ∥ L q L γ q ( R N ) L^q_\gamma(\mathbb{R}^N) L γ q ( R N ) f : R N → R f:\mathbb{R}^N\to\mathbb{R} f : R N → R ∣ ⋅ ∣ γ f ∈ L q ( R N ) |\cdot|^\gamma f\in L^q(\mathbb{R}^N) ∣ ⋅ ∣ γ f ∈ L q ( R N ) ∥ f ∥ L γ q = ∥ ∣ ⋅ ∣ γ f ∥ L q \|f\|_{L^q_\gamma}=\||\cdot|^\gamma f\|_{L^q} ∥ f ∥ L γ q = ∥∣ ⋅ ∣ γ f ∥ L q γ ∈ R \gamma\in\mathbb{R} γ ∈ R C 0 ( R N ) C_0(\mathbb{R}^N) C 0 ( R N ) C b ( R N ) C_b(\mathbb{R}^N) C b ( R N ) R N \mathbb{R}^N R N
By Fujita-type results [2 , 15 , 16 , 25 , 29 ] , when α ≤ 2 + l N \alpha\le\frac{2+l}{N} α ≤ N 2 + l α > 2 + l N \alpha>\frac{2+l}{N} α > N 2 + l α > 2 + l N \alpha>\frac{2+l}{N} α > N 2 + l u 0 u_0 u 0 l = 0 l=0 l = 0 [24 Theorem 3.2 (i), p. 372] that for α > 2 N \alpha>\frac{2}{N} α > N 2 C > 0 C>0 C > 0 u 0 ∈ C b ( R N ) u_0\in C_b(\mathbb{R}^N) u 0 ∈ C b ( R N ) u 0 ≥ 0 u_0\ge0 u 0 ≥ 0 lim inf ∣ x ∣ → ∞ ∣ x ∣ 2 α u 0 ( x ) > C \liminf_{|x|\to\infty}|x|^{\frac{2}{\alpha}}u_0(x)>C lim inf ∣ x ∣ → ∞ ∣ x ∣ α 2 u 0 ( x ) > C T max ( u 0 ) < ∞ T_{\max}(u_0)<\infty T m a x ( u 0 ) < ∞ α > 0 \alpha>0 α > 0 u 0 ∈ C b ( R N ) u_0\in C_b(\mathbb{R}^N) u 0 ∈ C b ( R N ) u 0 ≥ 0 u_0\ge0 u 0 ≥ 0 lim inf ∣ x ∣ → ∞ ∣ x ∣ γ u 0 ( x ) > 0 \liminf_{|x|\to\infty}|x|^\gamma u_0(x)>0 lim inf ∣ x ∣ → ∞ ∣ x ∣ γ u 0 ( x ) > 0 γ < 2 α \gamma<\frac{2}{\alpha} γ < α 2 T max ( u 0 ) < ∞ T_{\max}(u_0)<\infty T m a x ( u 0 ) < ∞ [24 ] . These results have been significantly refined in many works; see [26 , 30 , 31 , 35 ] . In particular, the authors of [35 ] extended the analysis to a broader class of initial data not covered by previous results. For instance, they showed that if u 0 ∈ C 0 ( R N ) u_0\in C_0(\mathbb{R}^N) u 0 ∈ C 0 ( R N ) u 0 ≥ 0 u_0\ge0 u 0 ≥ 0 u 0 ( x ) ≥ c ∣ x ∣ − 2 / α sin 2 ( log ∣ x ∣ ) u_0(x)\ge c|x|^{-2/\alpha}\sin^2(\log|x|) u 0 ( x ) ≥ c ∣ x ∣ − 2/ α sin 2 ( log ∣ x ∣ ) ∣ x ∣ |x| ∣ x ∣ c > 0 c>0 c > 0 T max ( u 0 ) < ∞ T_{\max}(u_0)<\infty T m a x ( u 0 ) < ∞ l ≠ 0 l\ne0 l = 0 − min { 2 , N } < l < 0 -\min\{2,N\}<l<0 − min { 2 , N } < l < 0 [4 Remark 1.3, p. 119] that if N ≥ 3 N\ge3 N ≥ 3 2 + l N < α < 2 + l 2 \frac{2+l}{N}<\alpha<\frac{2+l}{2} N 2 + l < α < 2 2 + l C > 0 C>0 C > 0 u 0 ∈ C 0 ( R N ) u_0\in C_0(\mathbb{R}^N) u 0 ∈ C 0 ( R N ) u 0 > 0 u_0>0 u 0 > 0 lim inf ∣ x ∣ → ∞ ∣ x ∣ 2 + l α u 0 ( x ) ≥ C \liminf_{|x|\to\infty}|x|^{\frac{2+l}{\alpha}}u_0(x)\ge C lim inf ∣ x ∣ → ∞ ∣ x ∣ α 2 + l u 0 ( x ) ≥ C T max ( u 0 ) < ∞ T_{\max}(u_0)<\infty T m a x ( u 0 ) < ∞ l ≥ 0 l\ge0 l ≥ 0 [37 Theorem 0.2’, p. 556] proved that for N ≥ 3 N\ge3 N ≥ 3 u 0 ∈ C b ( R N ) u_0\in C_b(\mathbb{R}^N) u 0 ∈ C b ( R N ) 0 ≤ u 0 ( x ) ≤ c ∣ x ∣ − l / α 0\le u_0(x)\le c|x|^{-l/\alpha} 0 ≤ u 0 ( x ) ≤ c ∣ x ∣ − l / α R N \mathbb{R}^N R N α ≥ α c ( l ) \alpha\ge\alpha_c(l) α ≥ α c ( l ) α c ( l ) \alpha_c(l) α c ( l ) [37 equation (0.7), p. 555] ,) and lim inf ∣ x ∣ → ∞ ∣ x ∣ 2 + l α u 0 ( x ) \liminf_{|x|\to\infty}|x|^{\frac{2+l}{\alpha}}u_0(x) lim inf ∣ x ∣ → ∞ ∣ x ∣ α 2 + l u 0 ( x ) [6 Theorem 5.1 (ii)] that, if l > − min { 2 , N } l>-\min\{2,N\} l > − min { 2 , N } α > 2 + l N \alpha>\frac{2+l}{N} α > N 2 + l 0 < γ < 2 + l α 0<\gamma<\frac{2+l}{\alpha} 0 < γ < α 2 + l u 0 ∈ C 0 ( R N ) u_0\in C_0(\mathbb{R}^N) u 0 ∈ C 0 ( R N ) u 0 ≥ 0 u_0\ge0 u 0 ≥ 0 lim inf ∣ x ∣ → ∞ ∣ x ∣ γ u 0 ( x ) > 0 \liminf_{|x|\to\infty}|x|^\gamma u_0(x)>0 lim inf ∣ x ∣ → ∞ ∣ x ∣ γ u 0 ( x ) > 0 T max ( u 0 ) < ∞ T_{\max}(u_0)<\infty T m a x ( u 0 ) < ∞
Throughout this paper, we let D λ D_\lambda D λ
( D λ f ) ( x ) = f ( λ x ) , ∀ λ > 0. (6) (D_\lambda f)(x)=f(\lambda x),\qquad\forall\ \lambda>0. \tag*{(6)} ( D λ f ) ( x ) = f ( λ x ) , ∀ λ > 0. (6) For − min { 2 , N } < l < N α -\min\{2,N\}<l<N\alpha − min { 2 , N } < l < N α
C l = ( 4 π ) − N 2 ∫ R N e − ∣ y ∣ 2 4 ∣ y ∣ − l α d y = ( e Δ ∣ ⋅ ∣ − l α ) ( 0 ) = 2 − l α Γ ( N 2 ) Γ ( N 2 − l 2 α ) . (7) C_l=(4\pi)^{-\frac{N}{2}}\int_{\mathbb{R}^N}e^{-\frac{|y|^2}{4}}|y|^{-\frac{l}{\alpha}}\,dy=\left(e^\Delta|\cdot|^{-\frac{l}{\alpha}}\right)(0)=\frac{2^{-\frac{l}{\alpha}}}{\Gamma\left(\frac{N}{2}\right)}\Gamma\left(\frac{N}{2}-\frac{l}{2\alpha}\right). \tag*{(7)} C l = ( 4 π ) − 2 N ∫ R N e − 4 ∣ y ∣ 2 ∣ y ∣ − α l d y = ( e Δ ∣ ⋅ ∣ − α l ) ( 0 ) = Γ ( 2 N ) 2 − α l Γ ( 2 N − 2 α l ) . (7) where Γ \Gamma Γ
Main results on blow-up criteria
Our main results on blow-up criteria, which extend from [35 Theorem 1.7, p. 146] to l ≠ 0 l \ne0 l = 0 l > 0 l > 0 l > 0 [21 , 18 , 19 , 20 , 13 ] .
Theorem 1.1 (Blow-up criterion for the Hénon equation). Let N ≥ 1 N \ge1 N ≥ 1 l l l α \alpha α γ \gamma γ
0 < l < N α , l α ≤ γ < N , 0 < α ≤ 2 + l γ . 0 < l < N\alpha,\qquad\frac{l}{\alpha} \le\gamma< N,\qquad0 < \alpha\le\frac{2+l}{\gamma}. 0 < l < N α , α l ≤ γ < N , 0 < α ≤ γ 2 + l . Let w ( x ) = ( 1 + ∣ x ∣ ) l α w(x)=(1+|x|)^{\frac{l}{\alpha}} w ( x ) = ( 1 + ∣ x ∣ ) α l C w ( R N ) = { ϕ ∈ C ( R N ) ; ∥ ϕ w ∥ L ∞ < ∞ } C_w(\mathbb{R}^N)=\{\phi\in C(\mathbb{R}^N);\ \|\phi w\|_{L^\infty}<\infty\} C w ( R N ) = { ϕ ∈ C ( R N ) ; ∥ ϕw ∥ L ∞ < ∞ } L w ∞ ( R N ) = { ϕ ; ∥ ϕ w ∥ L ∞ < ∞ } L^\infty_w(\mathbb{R}^N)=\{\phi;\ \|\phi w\|_{L^\infty}<\infty\} L w ∞ ( R N ) = { ϕ ; ∥ ϕw ∥ L ∞ < ∞ } u 0 ∈ C w ( R N ) u_0\in C_w(\mathbb{R}^N) u 0 ∈ C w ( R N ) u ∈ L l o c ∞ ( [ 0 , T max ( u 0 ) ) , L w ∞ ( R N ) ) u\in L^\infty_{\mathrm{loc}}([0,T_{\max}(u_0)),L^\infty_w(\mathbb{R}^N)) u ∈ L loc ∞ ([ 0 , T m a x ( u 0 )) , L w ∞ ( R N )) [37 Theorem 2.5, p. 565] , this solution is in fact a classical solution of (1.1)–(1.2).)
Let f ∈ C w ( R N ) f\in C_w(\mathbb{R}^N) f ∈ C w ( R N )
(i) 0 ≤ f ( x ) ≤ u 0 ( x ) 0\le f(x)\le u_0(x) 0 ≤ f ( x ) ≤ u 0 ( x ) x ∈ R N x\in\mathbb{R}^N x ∈ R N
(ii) there exists K > 0 K>0 K > 0
f ( x ) ≤ K ∣ x ∣ − γ for a.e. x ∈ R N . (8) f(x)\le K|x|^{-\gamma}\quad\text{for a.e. }x\in\mathbb{R}^N. \tag*{(8)} f ( x ) ≤ K ∣ x ∣ − γ for a.e. x ∈ R N . (8) Suppose there exist a sequence λ n → ∞ \lambda_n\to\infty λ n → ∞ z ∈ L l o c 1 ( R N ) z\in L^1_{\mathrm{loc}}(\mathbb{R}^N) z ∈ L loc 1 ( R N ) z ≢ 0 z\not\equiv0 z ≡ 0
λ n γ D λ n f → z in D ′ ( R N ) , as n → ∞ . (9) \lambda_n^\gamma D_{\lambda_n}f\to z\quad\text{in }\mathcal{D}'(\mathbb{R}^N),\quad\text{as }n\to\infty. \tag*{(9)} λ n γ D λ n f → z in D ′ ( R N ) , as n → ∞. (9) If α = 2 + l γ \alpha=\frac{2+l}{\gamma} α = γ 2 + l
∥ e z ∥ L ∞ > C l ( 2 + l 2 α ) 1 α , (10) \|e^{z}\|_{L^\infty}>C_l\left(\frac{2+l}{2\alpha}\right)^{\frac{1}{\alpha}}, \tag*{(10)} ∥ e z ∥ L ∞ > C l ( 2 α 2 + l ) α 1 , (10) where C l C_l C l T max ( u 0 ) < ∞ T_{\max}(u_0)<\infty T m a x ( u 0 ) < ∞
The second theorem is for the case l < 0 l < 0 l < 0
Theorem 1.2 (Blow-up criterion for the Hardy equation). Let N ≥ 1 N \ge1 N ≥ 1 l l l α \alpha α γ \gamma γ
− min { 2 , N } < l < 0 , 0 < γ < N , 0 < α ≤ 2 + l γ . -\min\{2,N\} < l < 0,\qquad0 < \gamma< N,\qquad0 < \alpha\le\frac{2+l}{\gamma}. − min { 2 , N } < l < 0 , 0 < γ < N , 0 < α ≤ γ 2 + l . Let u 0 ∈ C 0 ( R N ) u_0 \in C_0(\mathbb{R}^N) u 0 ∈ C 0 ( R N ) u ∈ C ( [ 0 , T max ( u 0 ) ) , C 0 ( R N ) ) u \in C([0,T_{\max}(u_0)),C_0(\mathbb{R}^N)) u ∈ C ([ 0 , T m a x ( u 0 )) , C 0 ( R N )) f ∈ C 0 ( R N ) f \in C_0(\mathbb{R}^N) f ∈ C 0 ( R N )
(i) 0 ≤ f ( x ) ≤ u 0 ( x ) 0 \le f(x) \le u_0(x) 0 ≤ f ( x ) ≤ u 0 ( x ) x ∈ R N x \in\mathbb{R}^N x ∈ R N
(ii) there exists K > 0 K > 0 K > 0
f ( x ) ≤ K ∣ x ∣ − γ for a.e. x ∈ R N . (11) f(x) \le K|x|^{-\gamma}\qquad\text{for a.e. }x \in\mathbb{R}^N. \tag*{(11)} f ( x ) ≤ K ∣ x ∣ − γ for a.e. x ∈ R N . (11) Assume there exist a sequence λ n → ∞ \lambda_n \to\infty λ n → ∞ z ∈ L l o c 1 ( R N ) z \in L^1_{\mathrm{loc}}(\mathbb{R}^N) z ∈ L loc 1 ( R N ) z ≢ 0 z \not\equiv0 z ≡ 0
λ n γ D λ n f → z in D ′ ( R N ) , as n → ∞ . (12) \lambda_n^\gamma D_{\lambda_n}f \to z\quad\text{in }D'(\mathbb{R}^N),\quad\text{as }n \to\infty. \tag*{(12)} λ n γ D λ n f → z in D ′ ( R N ) , as n → ∞. (12) If α = 2 + l γ \alpha= \frac{2+l}{\gamma} α = γ 2 + l
( e Δ z ) ( 0 ) > C l ( 2 + l 2 α ) 1 α , (13) (e^{\Delta}z)(0) > C_l\left(\frac{2+l}{2\alpha}\right)^{\frac{1}{\alpha}}, \tag*{(13)} ( e Δ z ) ( 0 ) > C l ( 2 α 2 + l ) α 1 , (13) where C l C_l C l T max ( u 0 ) < ∞ T_{\max}(u_0) < \infty T m a x ( u 0 ) < ∞
(vi) Suppose that the hypotheses of Theorem 1.1 or Theorem 1.2 are satisfied and that α < 2 + l γ \alpha< \frac{2+l}{\gamma} α < γ 2 + l
I γ = { ϕ ∈ C 0 ( R N ) ; lim inf ∣ x ∣ → ∞ ∣ x ∣ γ ϕ ( x ) > 0 } . I_\gamma=\{\phi\in C_0(\mathbb{R}^N);\ \liminf_{|x|\to\infty}|x|^\gamma\phi(x)>0\}. I γ = { ϕ ∈ C 0 ( R N ) ; ∣ x ∣ → ∞ lim inf ∣ x ∣ γ ϕ ( x ) > 0 } . It is easy to see that if u 0 ∈ I γ u_0\in I_\gamma u 0 ∈ I γ C > 0 C>0 C > 0 ρ > 0 \rho>0 ρ > 0 u 0 ( x ) ≥ C ∣ x ∣ − γ u_0(x)\ge C|x|^{-\gamma} u 0 ( x ) ≥ C ∣ x ∣ − γ ∣ x ∣ ≥ ρ |x|\ge\rho ∣ x ∣ ≥ ρ f ∈ C 0 ( R N ) f\in C_0(\mathbb{R}^N) f ∈ C 0 ( R N ) 0 ≤ f ≤ u 0 0\le f\le u_0 0 ≤ f ≤ u 0 f ( x ) = C ∣ x ∣ − γ f(x)=C|x|^{-\gamma} f ( x ) = C ∣ x ∣ − γ ∣ x ∣ ≥ ρ |x|\ge\rho ∣ x ∣ ≥ ρ ∣ x ∣ ≥ ρ λ |x|\ge\frac{\rho}{\lambda} ∣ x ∣ ≥ λ ρ λ γ D λ f ( x ) = C ∣ x ∣ − γ \lambda^\gamma D_\lambda f(x)=C|x|^{-\gamma} λ γ D λ f ( x ) = C ∣ x ∣ − γ λ γ D λ f → C ∣ ⋅ ∣ − γ \lambda^\gamma D_\lambda f\to C|\cdot|^{-\gamma} λ γ D λ f → C ∣ ⋅ ∣ − γ D ′ ( R N ) D'(\mathbb{R}^N) D ′ ( R N ) λ → ∞ \lambda\to\infty λ → ∞ Theorem 1.1 or Theorem 1.2 , as appropriate, implies that every solution with nonnegative initial data u 0 ∈ I γ u_0\in I_\gamma u 0 ∈ I γ [6 ] . Our results, however, apply to a strictly larger class of initial data. Indeed, even within C 0 ( R N ) C_0(\mathbb{R}^N) C 0 ( R N ) Theorem 1.1 or Theorem 1.2 that do not belong to I γ I_\gamma I γ u 0 ∈ C 0 ( R N ) u_0\in C_0(\mathbb{R}^N) u 0 ∈ C 0 ( R N ) ρ > 0 \rho>0 ρ > 0 u 0 ( x ) = ∣ x ∣ − γ sin 2 ( log ∣ x ∣ ) u_0(x)=|x|^{-\gamma}\sin^2(\log|x|) u 0 ( x ) = ∣ x ∣ − γ sin 2 ( log ∣ x ∣ ) ∣ x ∣ ≥ ρ |x|\ge\rho ∣ x ∣ ≥ ρ lim inf ∣ x ∣ → ∞ ∣ x ∣ γ u 0 ( x ) = 0 \liminf_{|x|\to\infty}|x|^\gamma u_0(x)=0 lim inf ∣ x ∣ → ∞ ∣ x ∣ γ u 0 ( x ) = 0 u 0 ∉ I γ u_0\notin I_\gamma u 0 ∈ / I γ f ∈ C 0 ( R N ) f\in C_0(\mathbb{R}^N) f ∈ C 0 ( R N ) 0 ≤ f ≤ u 0 0\le f\le u_0 0 ≤ f ≤ u 0 f ( x ) = ∣ x ∣ − γ sin 2 ( log ∣ x ∣ ) f(x)=|x|^{-\gamma}\sin^2(\log|x|) f ( x ) = ∣ x ∣ − γ sin 2 ( log ∣ x ∣ ) ∣ x ∣ ≥ ρ |x|\ge\rho ∣ x ∣ ≥ ρ λ γ D λ f ( x ) = ∣ x ∣ − γ sin 2 ( log ∣ x ∣ + log λ ) \lambda^\gamma D_\lambda f(x)=|x|^{-\gamma}\sin^2(\log|x|+\log\lambda) λ γ D λ f ( x ) = ∣ x ∣ − γ sin 2 ( log ∣ x ∣ + log λ ) ∣ x ∣ ≥ ρ λ |x|\ge\frac{\rho}{\lambda} ∣ x ∣ ≥ λ ρ λ n = e n π \lambda_n=e^{n\pi} λ n = e nπ u 0 u_0 u 0
(vii) Let α = 2 + l γ \alpha=\frac{2+l}{\gamma} α = γ 2 + l f ∈ C w ( R N ) f\in C_w(\mathbb{R}^N) f ∈ C w ( R N ) f ∈ C 0 ( R N ) f\in C_0(\mathbb{R}^N) f ∈ C 0 ( R N ) f ( x ) = c ∣ x ∣ − 2 + l α f(x)=c|x|^{-\frac{2+l}{\alpha}} f ( x ) = c ∣ x ∣ − α 2 + l ∣ x ∣ ≥ ρ > 0 |x|\ge\rho>0 ∣ x ∣ ≥ ρ > 0 c > 0 c>0 c > 0 ∣ x ∣ ≥ ρ λ |x|\ge\frac{\rho}{\lambda} ∣ x ∣ ≥ λ ρ λ 2 + l α D λ f ( x ) = c ∣ x ∣ − 2 + l α \lambda^{\frac{2+l}{\alpha}}D_\lambda f(x)=c|x|^{-\frac{2+l}{\alpha}} λ α 2 + l D λ f ( x ) = c ∣ x ∣ − α 2 + l λ 2 + l α D λ f → c ∣ ⋅ ∣ − 2 + l α \lambda^{\frac{2+l}{\alpha}}D_\lambda f\to c|\cdot|^{-\frac{2+l}{\alpha}} λ α 2 + l D λ f → c ∣ ⋅ ∣ − α 2 + l D ′ ( R N ) D'(\mathbb{R}^N) D ′ ( R N ) λ → ∞ \lambda\to\infty λ → ∞ u 0 ≥ f u_0\ge f u 0 ≥ f f ( x ) = c ∣ x ∣ − 2 + l α sin 2 ( log ∣ x ∣ ) f(x)=c|x|^{-\frac{2+l}{\alpha}}\sin^2(\log|x|) f ( x ) = c ∣ x ∣ − α 2 + l sin 2 ( log ∣ x ∣ ) ∣ x ∣ ≥ ρ |x|\ge\rho ∣ x ∣ ≥ ρ λ 2 + l α D λ f ( x ) = c ∣ x ∣ − 2 + l α sin 2 ( log ∣ x ∣ + log λ ) \lambda^{\frac{2+l}{\alpha}}D_\lambda f(x)=c|x|^{-\frac{2+l}{\alpha}}\sin^2(\log|x|+\log\lambda) λ α 2 + l D λ f ( x ) = c ∣ x ∣ − α 2 + l sin 2 ( log ∣ x ∣ + log λ ) ∣ x ∣ ≥ ρ λ |x|\ge\frac{\rho}{\lambda} ∣ x ∣ ≥ λ ρ λ n = e n π \lambda_n=e^{n\pi} λ n = e nπ u 0 ≥ f u_0\ge f u 0 ≥ f α = 2 + l γ \alpha=\frac{2+l}{\gamma} α = γ 2 + l [6 Theorem 5.1 (ii)] .
Main results on the upper bounds of the life-span.
We now turn to life-span results. Most of the existing results are related to the standard heat equation corresponding to l = 0 l=0 l = 0 T max ( λ φ ) T_{\max}(\lambda\varphi) T m a x ( λ φ ) φ \varphi φ λ → ∞ \lambda\to\infty λ → ∞ λ → 0 \lambda\to0 λ → 0 φ \varphi φ T max ( λ φ ) T_{\max}(\lambda\varphi) T m a x ( λ φ ) λ → 0 \lambda\to0 λ → 0 φ ∈ C b ( R N ) \varphi\in C_b(\mathbb{R}^N) φ ∈ C b ( R N ) φ \varphi φ [17 , 22 , 24 , 33 , 34 , 35 ] and the references therein. For the general case l > − min { 2 , N } l>-\min\{2,N\} l > − min { 2 , N } [28 ] considered two classes of initial data φ ∈ C b ( R N ) \varphi\in C_b(\mathbb{R}^N) φ ∈ C b ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0 φ \varphi φ φ \varphi φ c 1 , c 2 > 0 c_1,c_2>0 c 1 , c 2 > 0 c 1 ≤ φ ≤ c 2 c_1\le\varphi\le c_2 c 1 ≤ φ ≤ c 2 T max ( λ φ ) T_{\max}(\lambda\varphi) T m a x ( λ φ ) λ → 0 \lambda\to0 λ → 0 T max ( λ φ ) < ∞ T_{\max}(\lambda\varphi)<\infty T m a x ( λ φ ) < ∞ λ > 0 \lambda>0 λ > 0 l l l φ \varphi φ α \alpha α N N N [34 ] later established several lower bound estimates for the life-span for initial data belonging to Lebesgue and weighted Lebesgue spaces, as well as C 0 ( R N ) C_0(\mathbb{R}^N) C 0 ( R N )
Theorem 1.3 (Nonlinear Hardy parabolic equation [34 Theorem A.1] ). Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 − min { 2 , N } < l < 0 -\min\{2,N\} < l < 0 − min { 2 , N } < l < 0 φ ∈ C 0 ( R N ) \varphi\in C_{0}(\mathbb{R}^{N}) φ ∈ C 0 ( R N ) u ∈ C ( [ 0 , T max ( λ φ ) ) ; C 0 ( R N ) ) u \in C([0,T_{\max}(\lambda\varphi));C_{0}(\mathbb{R}^{N})) u ∈ C ([ 0 , T m a x ( λ φ )) ; C 0 ( R N )) λ φ \lambda\varphi λ φ λ > 0 \lambda> 0 λ > 0 C = C ( N , l , α ) > 0 C=C(N,l,\alpha)>0 C = C ( N , l , α ) > 0
T max ( λ φ ) ≥ C ( λ ∥ φ ∥ L ∞ ) − 2 α 2 + l , ∀ λ > 0. (14) T_{\max}(\lambda\varphi) \ge C(\lambda\|\varphi\|_{L^\infty})^{-\frac{2\alpha}{2+l}},
\qquad\forall\lambda>0. \tag*{(14)} T m a x ( λ φ ) ≥ C ( λ ∥ φ ∥ L ∞ ) − 2 + l 2 α , ∀ λ > 0. (14) Theorem 1.4 (Nonlinear Hénon parabolic equation [34 Theorem B.1 and Corollary B.2] ). Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 0 ≤ l < N α 0 \le l < N\alpha 0 ≤ l < N α φ ∈ L l α ∞ ( R N ) \varphi\in L^\infty_{\frac{l}{\alpha}}(\mathbb{R}^{N}) φ ∈ L α l ∞ ( R N ) u ∈ C ( ( 0 , T max ( λ φ ) ) ; L l α ∞ ( R N ) ) u \in C((0,T_{\max}(\lambda\varphi));L^\infty_{\frac{l}{\alpha}}(\mathbb{R}^{N})) u ∈ C (( 0 , T m a x ( λ φ )) ; L α l ∞ ( R N )) lim t → 0 ∥ u ( t ) − e t Δ ∥ L l α ∞ = 0 \lim_{t\to0}\|u(t)-e^{t\Delta}\|_{L^\infty_{\frac{l}{\alpha}}}=0 lim t → 0 ∥ u ( t ) − e t Δ ∥ L α l ∞ = 0 λ φ \lambda\varphi λ φ λ > 0 \lambda> 0 λ > 0 C = C ˉ ( N , l , α ) > 0 C=\bar C(N,l,\alpha)>0 C = C ˉ ( N , l , α ) > 0
T max ( λ φ ) ≥ C ( λ ∥ φ ∥ L l α ∞ ) − α , ∀ λ > 0. (15) T_{\max}(\lambda\varphi) \ge C\left(\lambda\|\varphi\|_{L^\infty_{\frac{l}{\alpha}}}\right)^{-\alpha},
\qquad\forall\lambda>0. \tag*{(15)} T m a x ( λ φ ) ≥ C ( λ ∥ φ ∥ L α l ∞ ) − α , ∀ λ > 0. (15) Theorem 1.5 (Nonlinear Hardy-Hénon parabolic equation [34 Theorem C.1 and Corollary C.2] ). Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 − min { 2 , N } < l < N α -\min\{2,N\} < l < N\alpha − min { 2 , N } < l < N α γ \gamma γ max { 0 , l α } < γ < min { N , 2 + l α } \max\{0,\frac{l}{\alpha}\}<\gamma<\min\{N,\frac{2+l}{\alpha}\} max { 0 , α l } < γ < min { N , α 2 + l } φ ∈ L l α ∞ ( R N ) \varphi\in L^\infty_{\frac{l}{\alpha}}(\mathbb{R}^{N}) φ ∈ L α l ∞ ( R N ) u ∈ C ( ( 0 , T max ( λ φ ) ) ; L l α ∞ ( R N ) ) ∩ C ( ( 0 , T max ( λ φ ) ) ; L γ ∞ ( R N ) ) u \in C((0,T_{\max}(\lambda\varphi));L^\infty_{\frac{l}{\alpha}}(\mathbb{R}^{N})) \cap C((0,T_{\max}(\lambda\varphi));L^\infty_\gamma(\mathbb{R}^{N})) u ∈ C (( 0 , T m a x ( λ φ )) ; L α l ∞ ( R N )) ∩ C (( 0 , T m a x ( λ φ )) ; L γ ∞ ( R N )) lim t → 0 ∥ u ( t ) − e t Δ ∥ L γ ∞ = 0 \lim_{t\to0}\|u(t)-e^{t\Delta}\|_{L^\infty_\gamma}=0 lim t → 0 ∥ u ( t ) − e t Δ ∥ L γ ∞ = 0 λ φ \lambda\varphi λ φ λ > 0 \lambda> 0 λ > 0 r r r ν \nu ν C = C ( N , l , α , γ ) > 0 C=C(N,l,\alpha,\gamma)>0 C = C ( N , l , α , γ ) > 0
T max ( λ φ ) ≥ C ( λ ∥ φ ∥ L γ ∞ ) − ( 2 + l 2 α − γ 2 ) − 1 , ∀ λ > 0. (16) T_{\max}(\lambda\varphi) \ge C\left(\lambda\|\varphi\|_{L^\infty_\gamma}\right)^{-\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}},
\qquad\forall\lambda>0. \tag*{(16)} T m a x ( λ φ ) ≥ C ( λ ∥ φ ∥ L γ ∞ ) − ( 2 α 2 + l − 2 γ ) − 1 , ∀ λ > 0. (16) We now study upper bounds for the life-span. The upper bounds results for the life-span with l ≠ 0 l\ne0 l = 0 [34 Proposition C.3, p. 621] , but only under the restrictive assumptions l > − min { 2 , N } l>-\min\{2,N\} l > − min { 2 , N } max { 0 , l N } < α < 2 + l N \max\{0,\frac{l}{N}\}<\alpha<\frac{2+l}{N} max { 0 , N l } < α < N 2 + l max { 0 , l α } < γ < N \max\{0,\frac{l}{\alpha}\}<\gamma<N max { 0 , α l } < γ < N [34 ] for (1.1)–(1.2), and allow us to analyze the asymptotic behavior of T max ( λ φ ) T_{\max}(\lambda\varphi) T m a x ( λ φ ) λ → ∞ \lambda\to\infty λ → ∞ λ → 0 \lambda\to0 λ → 0 [34 ] , and in contrast to many life-span estimates in the literature, our upper bounds not only characterize the dependence on λ \lambda λ φ \varphi φ
We distinguish between large and small values of λ > 0 \lambda>0 λ > 0 λ \lambda λ
Theorem 1.6. Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 0 < l < N α 0 < l < N\alpha 0 < l < N α φ ∈ L l α ∞ ( R N ) \varphi\in L^\infty_{\frac{l}{\alpha}}(\mathbb{R}^{N}) φ ∈ L α l ∞ ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0 φ ≢ 0 \varphi\not\equiv0 φ ≡ 0 T max ( λ φ ) < ∞ T_{\max}(\lambda\varphi)<\infty T m a x ( λ φ ) < ∞ λ > 0 \lambda>0 λ > 0
lim sup λ → ∞ λ α T max ( λ φ ) ≤ 1 α ( ∥ e Δ ⋅ ∣ ⋅ ∣ − l α ∥ L l α ∞ ∥ φ ∥ L l α ∞ ) α . \limsup_{\lambda\to\infty}\lambda^\alpha T_{\max}(\lambda\varphi)\le\frac{1}{\alpha}\left(\frac{\|e^\Delta\cdot|\cdot|^{-\frac{l}{\alpha}}\|_{L^\infty_{\frac{l}{\alpha}}}}{\|\varphi\|_{L^\infty_{\frac{l}{\alpha}}}}\right)^\alpha. λ → ∞ lim sup λ α T m a x ( λ φ ) ≤ α 1 ∥ φ ∥ L α l ∞ ∥ e Δ ⋅ ∣ ⋅ ∣ − α l ∥ L α l ∞ α . The equivalence (1.17) relies on the lower bound estimate (1.15), which is obtained from the well-posedness theory in L l ∞ ( R N ) L^\infty_l(\mathbb{R}^N) L l ∞ ( R N ) l > 0 l>0 l > 0 L ∞ ( R N ) L^\infty(\mathbb{R}^N) L ∞ ( R N ) N ≥ 1 N\ge1 N ≥ 1 α > 0 \alpha>0 α > 0 l > 0 l>0 l > 0 u 0 ∈ L ∞ ( R N ) u_0\in L^\infty(\mathbb{R}^N) u 0 ∈ L ∞ ( R N ) T ( u 0 ) > 0 T(u_0)>0 T ( u 0 ) > 0 u ∈ C ( ( 0 , T ( u 0 ) ] ; L ∞ ( R N ) ) u\in C((0,T(u_0)];L^\infty(\mathbb{R}^N)) u ∈ C (( 0 , T ( u 0 )] ; L ∞ ( R N )) u 0 ≡ 1 u_0\equiv1 u 0 ≡ 1 t ∈ ( 0 , T ( u 0 ) ] t\in(0,T(u_0)] t ∈ ( 0 , T ( u 0 )]
u ( t ) ≥ ∫ 0 t e ( t − σ ) Δ ∣ ⋅ ∣ l d σ . u(t)\ge\int_0^t e^{(t-\sigma)\Delta}|\cdot|^l\,d\sigma. u ( t ) ≥ ∫ 0 t e ( t − σ ) Δ ∣ ⋅ ∣ l d σ . Using the heat kernel representation and the reverse triangle inequality, we obtain
u ( t , x ) ≥ ( ∣ x ∣ − 1 ) l ∫ 0 t ∫ ∣ z ∣ ≤ 1 G ( t − σ , z ) d z d σ → ∞ , as ∣ x ∣ → ∞ , u(t,x)\ge(|x|-1)^l\int_0^t\int_{|z|\le1}G(t-\sigma,z)\,dz\,d\sigma\to\infty,\quad\text{as }|x|\to\infty, u ( t , x ) ≥ ( ∣ x ∣ − 1 ) l ∫ 0 t ∫ ∣ z ∣ ≤ 1 G ( t − σ , z ) d z d σ → ∞ , as ∣ x ∣ → ∞ , which contradicts the fact that u ( t ) ∈ L ∞ ( R N ) u(t)\in L^\infty(\mathbb{R}^N) u ( t ) ∈ L ∞ ( R N )
Theorem 1.7. Let N ≥ 1 N\ge1 N ≥ 1 α > 0 \alpha>0 α > 0 − min { 2 , N } < l < 0 -\min\{2,N\}<l<0 − min { 2 , N } < l < 0 C l C_l C l φ ∈ C 0 ( R N ) \varphi\in C_0(\mathbb{R}^N) φ ∈ C 0 ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0 φ ( 0 ) ≠ 0 \varphi(0)\ne0 φ ( 0 ) = 0 λ > 0 \lambda>0 λ > 0 u 0 = λ φ u_0=\lambda\varphi u 0 = λ φ T max ( λ φ ) < ∞ T_{\max}(\lambda\varphi)<\infty T m a x ( λ φ ) < ∞
lim sup λ → ∞ λ 2 α 2 + l T max ( λ φ ) ≤ ( 2 + l 2 α ) 2 2 + l ( C l φ ( 0 ) ) 2 α 2 + l . (18) \limsup_{\lambda\to\infty}\lambda^{\frac{2\alpha}{2+l}}T_{\max}(\lambda\varphi)\le\left(\frac{2+l}{2\alpha}\right)^{\frac{2}{2+l}}\left(\frac{C_l}{\varphi(0)}\right)^{\frac{2\alpha}{2+l}}. \tag*{(18)} λ → ∞ lim sup λ 2 + l 2 α T m a x ( λ φ ) ≤ ( 2 α 2 + l ) 2 + l 2 ( φ ( 0 ) C l ) 2 + l 2 α . (18) Remark 4. Let N ≥ 1 N\ge1 N ≥ 1 α > 0 \alpha>0 α > 0 − min { 2 , N } < l < 0 -\min\{2,N\}<l<0 − min { 2 , N } < l < 0 Theorem 1.3 and Theorem 1.7 , we obtain
T max ( λ φ ) ∼ λ − 2 α 2 + l , λ → ∞ , T_{\max}(\lambda\varphi)\sim\lambda^{-\frac{2\alpha}{2+l}},\quad\lambda\to\infty, T m a x ( λ φ ) ∼ λ − 2 + l 2 α , λ → ∞ , for all φ ∈ C 0 ( R N ) \varphi\in C_0(\mathbb{R}^N) φ ∈ C 0 ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0 φ ( 0 ) ≠ 0 \varphi(0)\ne0 φ ( 0 ) = 0
Theorem 1.8. Let N ≥ 1 N\ge1 N ≥ 1 α > 0 \alpha>0 α > 0 0 < l < N α 0<l<N\alpha 0 < l < N α C l C_l C l l α ≤ γ < min { N , 2 + l α } \frac{l}{\alpha}\le\gamma<\min\left\{N,\frac{2+l}{\alpha}\right\} α l ≤ γ < min { N , α 2 + l } ω ∈ L ∞ ( R N ) \omega\in L^\infty(\mathbb{R}^N) ω ∈ L ∞ ( R N ) 0 0 0 ω ≥ 0 \omega\ge0 ω ≥ 0 ω ≢ 0 \omega\not\equiv0 ω ≡ 0 φ ∈ L l o c 1 ( R N ) \varphi\in L^1_{\mathrm{loc}}(\mathbb{R}^N) φ ∈ L loc 1 ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0
φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≤ ε , \varphi(x)\ge\omega(x)|x|^{-\gamma},\quad\text{for }|x|\le\varepsilon, φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≤ ε , for some ε > 0 \varepsilon>0 ε > 0 T max ( λ φ ) < ∞ T_{\max}(\lambda\varphi)<\infty T m a x ( λ φ ) < ∞ λ > 0 \lambda>0 λ > 0
lim sup λ → ∞ λ ( 2 + l 2 α − γ 2 ) − 1 T max ( λ φ ) ≤ ( 2 + l 2 α ) ( 2 + l 2 α − γ 2 ) − 1 [ C l ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ ] ( 2 + l 2 α − γ 2 ) − 1 . \limsup_{\lambda\to\infty}\lambda^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}T_{\max}(\lambda\varphi)\le\left(\frac{2+l}{2\alpha}\right)^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}\left[\frac{C_l}{\|e^{\Delta}(\omega|\cdot|^{-\gamma})\|_{L^\infty}}\right]^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}. λ → ∞ lim sup λ ( 2 α 2 + l − 2 γ ) − 1 T m a x ( λ φ ) ≤ ( 2 α 2 + l ) ( 2 α 2 + l − 2 γ ) − 1 [ ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ C l ] ( 2 α 2 + l − 2 γ ) − 1 . Remark 5.
Let N ≥ 1 N\ge1 N ≥ 1 α > 0 \alpha>0 α > 0 0 < l < N α 0<l<N\alpha 0 < l < N α l α < γ < min { N , 2 + l α } \frac{l}{\alpha}<\gamma<\min\left\{N,\frac{2+l}{\alpha}\right\} α l < γ < min { N , α 2 + l } φ \varphi φ Theorem 1.8 such that φ ∈ L γ ∞ ( R N ) \varphi\in L^\infty_\gamma(\mathbb{R}^N) φ ∈ L γ ∞ ( R N ) Theorem 1.5 and Theorem 1.8 together show that
T max ( λ φ ) ∼ λ − ( 2 + l 2 α − γ 2 ) − 1 , λ → ∞ . T_{\max}(\lambda\varphi)\sim\lambda^{-\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}},\quad\lambda\to\infty. T m a x ( λ φ ) ∼ λ − ( 2 α 2 + l − 2 γ ) − 1 , λ → ∞. Let N ≥ 1 N\ge1 N ≥ 1 α > 0 \alpha>0 α > 0 0 < l < N α 0<l<N\alpha 0 < l < N α γ = l α \gamma=\frac{l}{\alpha} γ = α l φ \varphi φ Theorem 1.8 such that φ ∈ L l / α ∞ ( R N ) \varphi\in L^\infty_{l/\alpha}(\mathbb{R}^N) φ ∈ L l / α ∞ ( R N ) Theorem 1.4 and Theorem 1.8 together show that
T max ( λ φ ) ∼ λ − α , λ → ∞ , T_{\max}(\lambda\varphi)\sim\lambda^{-\alpha},\quad\lambda\to\infty, T m a x ( λ φ ) ∼ λ − α , λ → ∞ , in agreement with the equivalence (1.17).
Proposition C.3 in [34 ] provides an upper bound with the same rate in λ \lambda λ α < 2 + l N \alpha< \frac{2+l}{N} α < N 2 + l γ > l α \gamma> \frac{l}{\alpha} γ > α l
Theorem 1.9. Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 − min { 2 , N } < l < 0 -\min\{2,N\} < l < 0 − min { 2 , N } < l < 0 C l C_l C l 0 < γ < min { N , 2 + l α } 0 < \gamma< \min\{N,\frac{2+l}{\alpha}\} 0 < γ < min { N , α 2 + l } ω ∈ L ∞ ( R N ) \omega\in L^\infty(\mathbb{R}^N) ω ∈ L ∞ ( R N ) ω ≥ 0 \omega\ge0 ω ≥ 0 ω ≢ 0 \omega\not\equiv0 ω ≡ 0 φ ∈ L l o c 1 ( R N ) \varphi\in L^1_{\mathrm{loc}}(\mathbb{R}^N) φ ∈ L loc 1 ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0
φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≤ ε , \varphi(x) \ge\omega(x)|x|^{-\gamma}, \qquad\text{for } |x| \le\varepsilon, φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≤ ε , for some ε > 0 \varepsilon> 0 ε > 0 T max ( λ φ ) < ∞ T_{\max}(\lambda\varphi) < \infty T m a x ( λ φ ) < ∞ λ > 0 \lambda> 0 λ > 0
lim sup λ → ∞ λ ( 2 + l 2 α − γ 2 ) − 1 T max ( λ φ ) ≤ ( 2 + l 2 α ) ( 2 + l 2 α − α γ 2 α ) − 1 [ C l e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) ] ( 2 + l 2 α − γ 2 ) − 1 . \limsup_{\lambda\to\infty}\lambda^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}T_{\max}(\lambda\varphi)
\le\left(\frac{2+l}{2\alpha}\right)^{\left(\frac{2+l}{2\alpha}-\frac{\alpha\gamma}{2\alpha}\right)^{-1}}
\left[\frac{C_l}{e^{\Delta(\omega|\cdot|^{-\gamma})(0)}}\right]^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}. λ → ∞ lim sup λ ( 2 α 2 + l − 2 γ ) − 1 T m a x ( λ φ ) ≤ ( 2 α 2 + l ) ( 2 α 2 + l − 2 α α γ ) − 1 [ e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) C l ] ( 2 α 2 + l − 2 γ ) − 1 . We now turn to the case of small λ \lambda λ
Theorem 1.10. Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 0 < l < N α 0 < l < N\alpha 0 < l < N α C l C_l C l l α < γ < min { N , 2 + l α } \frac{l}{\alpha} < \gamma< \min\{N,\frac{2+l}{\alpha}\} α l < γ < min { N , α 2 + l } ω ∈ L ∞ ( R N ) \omega\in L^\infty(\mathbb{R}^N) ω ∈ L ∞ ( R N ) ω ≥ 0 \omega\ge0 ω ≥ 0 ω ≢ 0 \omega\not\equiv0 ω ≡ 0 φ ∈ L l o c 1 ( R N ) \varphi\in L^1_{\mathrm{loc}}(\mathbb{R}^N) φ ∈ L loc 1 ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0
φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≥ R , \varphi(x) \ge\omega(x)|x|^{-\gamma}, \qquad\text{for } |x| \ge R, φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≥ R , for some R > 0 R > 0 R > 0 T max ( λ φ ) < ∞ T_{\max}(\lambda\varphi) < \infty T m a x ( λ φ ) < ∞ λ > 0 \lambda> 0 λ > 0
lim sup λ → 0 λ ( 2 + l 2 α − γ 2 ) − 1 T max ( λ φ ) ≤ ( 2 + l 2 α ) ( 2 + l 2 α − α γ 2 α ) − 1 [ C l ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ ] ( 2 + l 2 α − γ 2 ) − 1 . \limsup_{\lambda\to0}\lambda^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}T_{\max}(\lambda\varphi)
\le\left(\frac{2+l}{2\alpha}\right)^{\left(\frac{2+l}{2\alpha}-\frac{\alpha\gamma}{2\alpha}\right)^{-1}}
\left[\frac{C_l}{\|e^{\Delta(\omega|\cdot|^{-\gamma})}\|_{L^\infty}}\right]^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}. λ → 0 lim sup λ ( 2 α 2 + l − 2 γ ) − 1 T m a x ( λ φ ) ≤ ( 2 α 2 + l ) ( 2 α 2 + l − 2 α α γ ) − 1 [ ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ C l ] ( 2 α 2 + l − 2 γ ) − 1 . Theorem 1.11. Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 − min { 2 , N } < l < 0 -\min\{2,N\} < l < 0 − min { 2 , N } < l < 0 C l C_l C l (7) . Let 0 < γ < min { N , 2 + l α } 0 < \gamma< \min\{N,\frac{2+l}{\alpha}\} 0 < γ < min { N , α 2 + l } ω ∈ L ∞ ( R N ) \omega\in L^\infty(\mathbb{R}^N) ω ∈ L ∞ ( R N ) 0 0 0 ω ≥ 0 \omega\ge0 ω ≥ 0 ω ≢ 0 \omega\not\equiv0 ω ≡ 0 φ ∈ L l o c 1 ( R N ) \varphi\in L^1_{\mathrm{loc}}(\mathbb{R}^N) φ ∈ L loc 1 ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0
φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≥ R , \varphi(x) \ge\omega(x)|x|^{-\gamma}, \qquad\text{for } |x| \ge R, φ ( x ) ≥ ω ( x ) ∣ x ∣ − γ , for ∣ x ∣ ≥ R , for some R > 0 R > 0 R > 0 T max ( λ φ ) < ∞ T_{\max}(\lambda\varphi) < \infty T m a x ( λ φ ) < ∞ λ > 0 \lambda> 0 λ > 0
lim sup λ → 0 λ ( 2 + l 2 α − γ 2 ) − 1 T max ( λ φ ) ≤ ( 2 + l 2 α ) ( 2 + l 2 α − γ 2 ) − 1 [ C l e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) ] ( 2 + l 2 α − γ 2 ) − 1 . \limsup_{\lambda\to0} \lambda^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}} T_{\max}(\lambda\varphi) \le\left(\frac{2+l}{2\alpha}\right)^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}} \left[\frac{C_l}{e^{\Delta(\omega|\cdot|^{-\gamma})(0)}}\right]^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}. λ → 0 lim sup λ ( 2 α 2 + l − 2 γ ) − 1 T m a x ( λ φ ) ≤ ( 2 α 2 + l ) ( 2 α 2 + l − 2 γ ) − 1 [ e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) C l ] ( 2 α 2 + l − 2 γ ) − 1 . Remark 8.
Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 − min { 2 , N } < l < 0 -\min\{2,N\} < l < 0 − min { 2 , N } < l < 0 0 < γ < min { N , 2 + l α } 0 < \gamma< \min\{N,\frac{2+l}{\alpha}\} 0 < γ < min { N , α 2 + l } φ \varphi φ Theorem 1.11 be such that φ ∈ L ∞ ( R N ) \varphi\in L^\infty(\mathbb{R}^N) φ ∈ L ∞ ( R N ) Theorems 1.5 and 1.11 together imply that
T max ( λ φ ) ∼ λ − ( 2 + l 2 α − γ 2 ) − 1 , λ → 0. T_{\max}(\lambda\varphi) \sim\lambda^{-\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}, \qquad\lambda\to0. T m a x ( λ φ ) ∼ λ − ( 2 α 2 + l − 2 γ ) − 1 , λ → 0. Proposition C.3 in [34 ] provides an upper bound with the same rate for λ \lambda λ
In [6 Remark 5.2] , the authors showed that λ ( 2 + l 2 α − γ 2 ) − 1 T max ( λ φ ) ≤ C \lambda^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}} T_{\max}(\lambda\varphi) \le C λ ( 2 α 2 + l − 2 γ ) − 1 T m a x ( λ φ ) ≤ C λ > 0 \lambda> 0 λ > 0 φ = ( 1 + ∣ ⋅ ∣ 2 ) − γ 2 \varphi= (1+|\cdot|^2)^{-\frac{\gamma}{2}} φ = ( 1 + ∣ ⋅ ∣ 2 ) − 2 γ C > 0 C > 0 C > 0
In proving our results, we draw on the ideas of [6 ] , [34 ] , and [35 ] . However, these arguments cannot be directly adapted to our setting, and new techniques are needed to establish the blow-up results and derive the upper bound estimates.
The remainder of the paper is organized as follows. In § 2 , we establish the necessary conditions for the local existence. In § 3 , we prove the blow-up criteria stated in Theorems 1.1 and 1.2 . Finally, § 4 is devoted to establishing the upper bound estimates stated in Theorems 1.6 –1.11 .
Throughout this paper, for positive functions f f f g g g f ( x ) ∼ g ( x ) f(x) \sim g(x) f ( x ) ∼ g ( x ) x → x 0 x \to x_0 x → x 0 C 1 C_1 C 1 C 2 C_2 C 2 C 1 g ( x ) ≤ f ( x ) ≤ C 2 g ( x ) C_1g(x) \le f(x) \le C_2g(x) C 1 g ( x ) ≤ f ( x ) ≤ C 2 g ( x ) x x x x 0 x_0 x 0
Necessary conditions for the existence
In this section, we give the necessary conditions for the existence of a nonnegative solution to (3) . Similar conditions were obtained in [38 Theorem 1, p. 546] for the case l = 0 l = 0 l = 0 [1 Theorem 2.12, p. 27] .
Lemma 2.1 (A Reverse Hölder inequality). Let Ω \Omega Ω R N \mathbb{R}^N R N p ∈ ( 1 , ∞ ) p \in(1,\infty) p ∈ ( 1 , ∞ ) f , g ∈ L 1 ( Ω ) f,g \in L^1(\Omega) f , g ∈ L 1 ( Ω ) ∣ g ∣ > 0 |g| > 0 ∣ g ∣ > 0 Ω \Omega Ω
∫ Ω ∣ f ( y ) ∣ p ∣ g ( y ) ∣ 1 − p d y ≥ ( ∫ Ω ∣ f ( y ) ∣ d y ) p ( ∫ Ω ∣ g ( y ) ∣ d y ) 1 − p . \int_\Omega|f(y)|^p|g(y)|^{1-p}\,dy \ge\left(\int_\Omega|f(y)|\,dy\right)^p \left(\int_\Omega|g(y)|\,dy\right)^{1-p}. ∫ Ω ∣ f ( y ) ∣ p ∣ g ( y ) ∣ 1 − p d y ≥ ( ∫ Ω ∣ f ( y ) ∣ d y ) p ( ∫ Ω ∣ g ( y ) ∣ d y ) 1 − p . We also recall the commutation relation between D λ D_\lambda D λ R N \mathbb{R}^N R N
D λ e τ λ 2 Δ = e τ Δ D λ , ∀ λ > 0 , τ > 0. (19) D_\lambda e^{\tau\lambda^2\Delta} = e^{\tau\Delta}D_\lambda,\qquad\forall\lambda> 0,\quad\tau> 0. \tag*{(19)} D λ e τ λ 2 Δ = e τ Δ D λ , ∀ λ > 0 , τ > 0. (19) In particular, we have
D t e t Δ = e Δ D t , ∀ t > 0. (20) D_{\sqrt{t}}e^{t\Delta} = e^\Delta D_{\sqrt{t}},\qquad\forall t > 0. \tag*{(20)} D t e t Δ = e Δ D t , ∀ t > 0. (20) We have the following result.
Proposition 2.2. Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 0 ≤ l < N α 0 \le l < N\alpha 0 ≤ l < N α C l C_l C l u 0 ≥ 0 u_0 \ge0 u 0 ≥ 0 R N \mathbb{R}^N R N T max ( u 0 ) T_{\max}(u_0) T m a x ( u 0 ) u u u ( 0 , T max ( u 0 ) ) × R N (0,T_{\max}(u_0)) \times\mathbb{R}^N ( 0 , T m a x ( u 0 )) × R N
(a) for all ( t , x ) ∈ ( 0 , T max ( u 0 ) ) × R N (t,x) \in(0,T_{\max}(u_0)) \times\mathbb{R}^N ( t , x ) ∈ ( 0 , T m a x ( u 0 )) × R N
2 2 + l α t 2 + l 2 [ ( e t Δ u 0 ) ( x ) ] α ≤ C l α . (21) \frac{2}{2+l}\alpha t^{\frac{2+l}{2}}\left[(e^{t\Delta}u_0)(x)\right]^\alpha\le C_l^\alpha. \tag*{(21)} 2 + l 2 α t 2 2 + l [ ( e t Δ u 0 ) ( x ) ] α ≤ C l α . (21) Consequently,
2 2 + l α t 2 + l 2 ∥ e t Δ u 0 ∥ L ∞ α ≤ C l α , (22) \frac{2}{2+l}\alpha t^{\frac{2+l}{2}}\|e^{t\Delta}u_0\|_{L^\infty}^\alpha\le C_l^\alpha, \tag*{(22)} 2 + l 2 α t 2 2 + l ∥ e t Δ u 0 ∥ L ∞ α ≤ C l α , (22) for all t ∈ ( 0 , T max ( u 0 ) ) t \in(0,T_{\max}(u_0)) t ∈ ( 0 , T m a x ( u 0 ))
(b) If 0 < T max ( u 0 ) < ∞ 0 < T_{\max}(u_0) < \infty 0 < T m a x ( u 0 ) < ∞
2 2 + l α ( T max ( u 0 ) ) 2 + l 2 ∥ e T max ( u 0 ) Δ u 0 ∥ L ∞ α ≤ C l α . (23) \frac{2}{2+l}\alpha\left(T_{\max}(u_0)\right)^{\frac{2+l}{2}}\|e^{T_{\max}(u_0)\Delta}u_0\|_{L^\infty}^\alpha\le C_l^\alpha. \tag*{(23)} 2 + l 2 α ( T m a x ( u 0 ) ) 2 2 + l ∥ e T m a x ( u 0 ) Δ u 0 ∥ L ∞ α ≤ C l α . (23) (c) If 0 < T max ( u 0 ) < ∞ 0 < T_{\max}(u_0) < \infty 0 < T m a x ( u 0 ) < ∞ u 0 = λ φ u_0 = \lambda\varphi u 0 = λ φ λ > 0 \lambda> 0 λ > 0
λ α 2 2 + l α ( T max ( λ φ ) ) 2 + l 2 ∥ e T max ( λ φ ) Δ φ ∥ L ∞ α ≤ C l α . (24) \lambda^\alpha\frac{2}{2+l}\alpha\left(T_{\max}(\lambda\varphi)\right)^{\frac{2+l}{2}}\|e^{T_{\max}(\lambda\varphi)\Delta}\varphi\|_{L^\infty}^\alpha\le C_l^\alpha. \tag*{(24)} λ α 2 + l 2 α ( T m a x ( λ φ ) ) 2 2 + l ∥ e T m a x ( λ φ ) Δ φ ∥ L ∞ α ≤ C l α . (24) Proof. (a) The case l = 0 l = 0 l = 0 [38 Theorem 1, p. 546] , whereas the case 0 < l < N α 0 < l < N\alpha 0 < l < N α [6 ] . See also [36 Lemma 4.1] .
(b) Since inequality (2.3) is valid for every 0 < t < T max ( u 0 ) 0 < t < T_{\max}(u_0) 0 < t < T m a x ( u 0 ) t = T max ( u 0 ) t = T_{\max}(u_0) t = T m a x ( u 0 )
(c) Inequality (2.6) is an immediate consequence of (2.5).
In the Hénon case we have the following result using weighted spaces.
Proposition 2.3. Let N ≥ 1 N \ge1 N ≥ 1 α > 0 \alpha> 0 α > 0 0 ≤ l < N α 0 \le l < N\alpha 0 ≤ l < N α u 0 ≥ 0 u_0 \ge0 u 0 ≥ 0 R N \mathbb{R}^N R N T max ( u 0 ) T_{\max}(u_0) T m a x ( u 0 ) u u u ( 0 , T max ( u 0 ) ) × R N (0,T_{\max}(u_0)) \times\mathbb{R}^N ( 0 , T m a x ( u 0 )) × R N
(a) for every ( t , x ) ∈ ( 0 , T max ( u 0 ) ) × R N (t,x) \in(0,T_{\max}(u_0)) \times\mathbb{R}^N ( t , x ) ∈ ( 0 , T m a x ( u 0 )) × R N
α t [ ∣ x ∣ l α ( e t Δ u 0 ) ( x ) ] α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − l α ∥ L l α ∞ α . (25) \alpha t\left[|x|^{\frac{l}{\alpha}}(e^{t\Delta}u_0)(x)\right]^\alpha\le\|e^\Delta\cdot|\cdot|^{-\frac{l}{\alpha}}\|_{L^\infty_{\frac{l}{\alpha}}}^\alpha. \tag*{(25)} α t [ ∣ x ∣ α l ( e t Δ u 0 ) ( x ) ] α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − α l ∥ L α l ∞ α . (25) Therefore, for each t ∈ ( 0 , T max ( u 0 ) ) t \in(0,T_{\max}(u_0)) t ∈ ( 0 , T m a x ( u 0 ))
α t ∥ e t Δ u 0 ∥ L l α ∞ α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − l α ∥ L l α ∞ α . (26) \alpha t\|e^{t\Delta}u_0\|_{L^\infty_{\frac{l}{\alpha}}}^\alpha\le\|e^\Delta\cdot|\cdot|^{-\frac{l}{\alpha}}\|_{L^\infty_{\frac{l}{\alpha}}}^\alpha. \tag*{(26)} α t ∥ e t Δ u 0 ∥ L α l ∞ α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − α l ∥ L α l ∞ α . (26) (b) If 0 < T max ( u 0 ) < ∞ 0 < T_{\max}(u_0) < \infty 0 < T m a x ( u 0 ) < ∞
α T max ( u 0 ) ∥ e T max ( u 0 ) Δ u 0 ∥ L l α ∞ α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − l α ∥ L l α ∞ α . (27) \alpha T_{\max}(u_0)\|e^{T_{\max}(u_0)\Delta}u_0\|_{L^\infty_{\frac{l}{\alpha}}}^\alpha\le\|e^\Delta\cdot|\cdot|^{-\frac{l}{\alpha}}\|_{L^\infty_{\frac{l}{\alpha}}}^\alpha. \tag*{(27)} α T m a x ( u 0 ) ∥ e T m a x ( u 0 ) Δ u 0 ∥ L α l ∞ α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − α l ∥ L α l ∞ α . (27) (c) If 0 < T max ( u 0 ) < ∞ 0 < T_{\max}(u_0) < \infty 0 < T m a x ( u 0 ) < ∞ u 0 = λ φ u_0 = \lambda\varphi u 0 = λ φ λ > 0 \lambda> 0 λ > 0
λ α α T max ( λ φ ) ∥ e T max ( λ φ ) Δ φ ∥ L l α ∞ α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − l α ∥ L l α ∞ α . (28) \lambda^\alpha\alpha T_{\max}(\lambda\varphi)\|e^{T_{\max}(\lambda\varphi)\Delta}\varphi\|_{L^\infty_{\frac{l}{\alpha}}}^\alpha\le\|e^\Delta\cdot|\cdot|^{-\frac{l}{\alpha}}\|_{L^\infty_{\frac{l}{\alpha}}}^\alpha. \tag*{(28)} λ α α T m a x ( λ φ ) ∥ e T m a x ( λ φ ) Δ φ ∥ L α l ∞ α ≤ ∥ e Δ ⋅ ∣ ⋅ ∣ − α l ∥ L α l ∞ α . (28) Proof. (a) The proof relies on several arguments from [6 , 33 , 38 ] . Without loss of generality, we assume that u 0 ≠ 0 u_0 \ne0 u 0 = 0 0 < l < N α 0 < l < N\alpha 0 < l < N α τ ∈ ( 0 , T max ( u 0 ) ) \tau\in(0,T_{\max}(u_0)) τ ∈ ( 0 , T m a x ( u 0 )) 0 < t < τ 0 < t < \tau 0 < t < τ
e ( τ − t ) Δ u ( t ) = e τ Δ u 0 + ∫ 0 t e ( τ − σ ) Δ [ ∣ ⋅ ∣ l u α + 1 ( σ ) ] d σ . (29) e^{(\tau-t)\Delta}u(t)=e^{\tau\Delta}u_0+\int_0^t e^{(\tau-\sigma)\Delta}\left[|\cdot|^{l}u^{\alpha+1}(\sigma)\right]d\sigma. \tag*{(29)} e ( τ − t ) Δ u ( t ) = e τ Δ u 0 + ∫ 0 t e ( τ − σ ) Δ [ ∣ ⋅ ∣ l u α + 1 ( σ ) ] d σ . (29) Using Lemma 2.1 (A reverse Hölder inequality) with f ( y ) = G ( τ − σ , x − y ) u ( σ , y ) f(y)=G(\tau-\sigma,x-y)u(\sigma,y) f ( y ) = G ( τ − σ , x − y ) u ( σ , y ) g ( y ) = G ( τ − σ , x − y ) ∣ y ∣ − l α g(y)=G(\tau-\sigma,x-y)|y|^{-\frac{l}{\alpha}} g ( y ) = G ( τ − σ , x − y ) ∣ y ∣ − α l p = α + 1 p=\alpha+1 p = α + 1
e ( τ − σ ) Δ [ ∣ ⋅ ∣ l u α + 1 ( σ ) ] ( x ) = ∫ R N G ( τ − σ , x − y ) ∣ y ∣ l u α + 1 ( σ , y ) d y = ∫ R N ( G ( τ − σ , x − y ) ∣ y ∣ − l α ) − α × ( G ( τ − σ , x − y ) u ( σ , y ) ) α + 1 d y ≥ ( ∫ R N G ( τ − σ , x − y ) ∣ y ∣ − l α d y ) − α × ( ∫ R N G ( τ − σ , x − y ) u ( σ , y ) d y ) α + 1 = ( e ( τ − σ ) Δ [ ∣ ⋅ ∣ − l α ] ( x ) ) − α ( e ( τ − σ ) Δ u ( σ ) ) α + 1 ( x ) , (30) \begin{aligned}
e^{(\tau-\sigma)\Delta}\left[|\cdot|^{l}u^{\alpha+1}(\sigma)\right](x)
&=\int_{\mathbb{R}^N}G(\tau-\sigma,x-y)|y|^{l}u^{\alpha+1}(\sigma,y)dy\\
&=\int_{\mathbb{R}^N}\left(G(\tau-\sigma,x-y)|y|^{-\frac{l}{\alpha}}\right)^{-\alpha}\\
&\qquad\qquad\times\left(G(\tau-\sigma,x-y)u(\sigma,y)\right)^{\alpha+1}dy\\
&\ge\left(\int_{\mathbb{R}^N}G(\tau-\sigma,x-y)|y|^{-\frac{l}{\alpha}}dy\right)^{-\alpha}\\
&\qquad\qquad\times\left(\int_{\mathbb{R}^N}G(\tau-\sigma,x-y)u(\sigma,y)dy\right)^{\alpha+1}\\
&=\left(e^{(\tau-\sigma)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right](x)\right)^{-\alpha}\left(e^{(\tau-\sigma)\Delta}u(\sigma)\right)^{\alpha+1}(x), \tag*{(30)}
\end{aligned} e ( τ − σ ) Δ [ ∣ ⋅ ∣ l u α + 1 ( σ ) ] ( x ) = ∫ R N G ( τ − σ , x − y ) ∣ y ∣ l u α + 1 ( σ , y ) d y = ∫ R N ( G ( τ − σ , x − y ) ∣ y ∣ − α l ) − α × ( G ( τ − σ , x − y ) u ( σ , y ) ) α + 1 d y ≥ ( ∫ R N G ( τ − σ , x − y ) ∣ y ∣ − α l d y ) − α × ( ∫ R N G ( τ − σ , x − y ) u ( σ , y ) d y ) α + 1 = ( e ( τ − σ ) Δ [ ∣ ⋅ ∣ − α l ] ( x ) ) − α ( e ( τ − σ ) Δ u ( σ ) ) α + 1 ( x ) , (30) for all x ∈ R N x \in\mathbb{R}^N x ∈ R N 0 < σ < τ 0 < \sigma< \tau 0 < σ < τ (29) and (30) , we get
e ( τ − t ) Δ u ( t ) ≥ e τ Δ u 0 + ∫ 0 t ( e ( τ − σ ) Δ [ ∣ ⋅ ∣ − l α ] ) − α ( e ( τ − σ ) Δ u ( σ ) ) α + 1 d σ : = Φ ( t ) . e^{(\tau-t)\Delta}u(t)\ge e^{\tau\Delta}u_0+\int_0^t\left(e^{(\tau-\sigma)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right)^{-\alpha}\left(e^{(\tau-\sigma)\Delta}u(\sigma)\right)^{\alpha+1}d\sigma:=\Phi(t). e ( τ − t ) Δ u ( t ) ≥ e τ Δ u 0 + ∫ 0 t ( e ( τ − σ ) Δ [ ∣ ⋅ ∣ − α l ] ) − α ( e ( τ − σ ) Δ u ( σ ) ) α + 1 d σ := Φ ( t ) . We have,
Φ ′ ( t ) = ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − l α ] ) − α ( e ( τ − t ) Δ u ( t ) ) α + 1 ≥ ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − l α ] ) − α ( Φ ( t ) ) α + 1 . \begin{aligned}
\Phi'(t)=\left(e^{(\tau-t)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right)^{-\alpha}\left(e^{(\tau-t)\Delta}u(t)\right)^{\alpha+1} \\
&\ge\left(e^{(\tau-t)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right)^{-\alpha}\left(\Phi(t)\right)^{\alpha+1}.
\end{aligned} Φ ′ ( t ) = ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − α l ] ) − α ( e ( τ − t ) Δ u ( t ) ) α + 1 ≥ ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − α l ] ) − α ( Φ ( t ) ) α + 1 . Therefore,
( Φ − α ( t ) ) ′ = − α Φ ′ ( t ) ( Φ ( t ) ) − α − 1 ≤ − α ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − l α ] ) − α . \left(\Phi^{-\alpha}(t)\right)'=-\alpha\Phi'(t)\left(\Phi(t)\right)^{-\alpha-1}\le-\alpha\left(e^{(\tau-t)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right)^{-\alpha}. ( Φ − α ( t ) ) ′ = − α Φ ′ ( t ) ( Φ ( t ) ) − α − 1 ≤ − α ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − α l ] ) − α . By integrating both sides of the above inequality over ( 0 , τ ) (0,\tau) ( 0 , τ )
Φ − α ( τ ) − Φ − α ( 0 ) ≤ − α ∫ 0 τ ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − l α ] ) − α d t . (31) \Phi^{-\alpha}(\tau)-\Phi^{-\alpha}(0)\le-\alpha\int_0^\tau\left(e^{(\tau-t)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right)^{-\alpha}dt. \tag*{(31)} Φ − α ( τ ) − Φ − α ( 0 ) ≤ − α ∫ 0 τ ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − α l ] ) − α d t . (31) Since Φ ( 0 ) = e τ Δ u 0 \Phi(0)=e^{\tau\Delta}u_0 Φ ( 0 ) = e τ Δ u 0 Φ ( τ ) > 0 \Phi(\tau)>0 Φ ( τ ) > 0 (31) that
( e τ Δ u 0 ) α ∫ 0 τ ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − l α ] ) − α d t ≤ 1 α . (32) \left(e^{\tau\Delta}u_0\right)^\alpha\int_0^\tau\left(e^{(\tau-t)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right)^{-\alpha}dt\le\frac{1}{\alpha}. \tag*{(32)} ( e τ Δ u 0 ) α ∫ 0 τ ( e ( τ − t ) Δ [ ∣ ⋅ ∣ − α l ] ) − α d t ≤ α 1 . (32) Inequality (32) can be rewritten as
( ∣ ⋅ ∣ − l α e τ Δ u 0 ) α ∫ 0 τ ( ∣ ⋅ ∣ l α e ( τ − t ) Δ [ ∣ ⋅ ∣ − l α ] ) − α d t ≤ 1 α . (33) \left(|\cdot|^{-\frac{l}{\alpha}}e^{\tau\Delta}u_0\right)^\alpha\int_0^\tau\left(|\cdot|^{\frac{l}{\alpha}}e^{(\tau-t)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right)^{-\alpha}dt\le\frac{1}{\alpha}. \tag*{(33)} ( ∣ ⋅ ∣ − α l e τ Δ u 0 ) α ∫ 0 τ ( ∣ ⋅ ∣ α l e ( τ − t ) Δ [ ∣ ⋅ ∣ − α l ] ) − α d t ≤ α 1 . (33) Consequently, (33) yields
( ∣ ⋅ ∣ l α e τ Δ u 0 ) α ∫ 0 τ ∥ e ( τ − t ) Δ [ ∣ ⋅ ∣ − l α ] ∥ L l α ∞ − α d t ≤ 1 α . (34) \left(|\cdot|^{\frac{l}{\alpha}}e^{\tau\Delta}u_0\right)^\alpha\int_0^\tau\left\|e^{(\tau-t)\Delta}\left[|\cdot|^{-\frac{l}{\alpha}}\right]\right\|_{L^\infty_{\frac{l}{\alpha}}}^{-\alpha}dt\le\frac{1}{\alpha}. \tag*{(34)} ( ∣ ⋅ ∣ α l e τ Δ u 0 ) α ∫ 0 τ e ( τ − t ) Δ [ ∣ ⋅ ∣ − α l ] L α l ∞ − α d t ≤ α 1 . (34) From (2.2) and the homogeneity of ∣ ⋅ ∣ − l α |\cdot|^{-\frac{l}{\alpha}} ∣ ⋅ ∣ − α l
∥ e Δ ∣ ⋅ ∣ − l α ∥ L l α ∞ = ∥ D t e t Δ D 1 t ∣ ⋅ ∣ − l α ∥ L l α ∞ = t − l 2 α ∥ e t Δ D 1 t ∣ ⋅ ∣ − l α ∥ L l α ∞ = ∥ e t Δ ∣ ⋅ ∣ − l α ∥ L l α ∞ , ∀ t > 0. (35) \begin{aligned}
\left\|e^{\Delta}|\cdot|^{-\frac{l}{\alpha}}\right\|_{L^\infty_{\frac{l}{\alpha}}}
=
\left\|D_{\sqrt{t}}e^{t\Delta}D_{\frac{1}{\sqrt{t}}}|\cdot|^{-\frac{l}{\alpha}}\right\|_{L^\infty_{\frac{l}{\alpha}}}
=
t^{-\frac{l}{2\alpha}}\left\|e^{t\Delta}D_{\frac{1}{\sqrt{t}}}|\cdot|^{-\frac{l}{\alpha}}\right\|_{L^\infty_{\frac{l}{\alpha}}} \\
&=
\left\|e^{t\Delta}|\cdot|^{-\frac{l}{\alpha}}\right\|_{L^\infty_{\frac{l}{\alpha}}},\qquad\forall t>0. \tag*{(35)}
\end{aligned} e Δ ∣ ⋅ ∣ − α l L α l ∞ = D t e t Δ D t 1 ∣ ⋅ ∣ − α l L α l ∞ = t − 2 α l e t Δ D t 1 ∣ ⋅ ∣ − α l L α l ∞ = e t Δ ∣ ⋅ ∣ − α l L α l ∞ , ∀ t > 0. (35) Therefore, (2.7) follows from (2.16) and (2.17).
(b) Since inequality (2.7) holds for every 0 < t < T max ( u 0 ) 0<t<T_{\max}(u_0) 0 < t < T m a x ( u 0 ) t = T max ( u 0 ) t=T_{\max}(u_0) t = T m a x ( u 0 )
(c) (2.10) follows immediately from (2.9).
Remark 9. For 0 < l < N α 0<l<N\alpha 0 < l < N α [10 ] implies that ∥ e Δ ∣ ⋅ ∣ − l α ∥ L l α ∞ ≤ C ∥ ∣ ⋅ ∣ − l α ∥ L l α ∞ = C \left\|e^\Delta|\cdot|^{-\frac{l}{\alpha}}\right\|_{L^\infty_{\frac{l}{\alpha}}}\leq C\left\||\cdot|^{-\frac{l}{\alpha}}\right\|_{L^\infty_{\frac{l}{\alpha}}}=C e Δ ∣ ⋅ ∣ − α l L α l ∞ ≤ C ∣ ⋅ ∣ − α l L α l ∞ = C
Proposition 2.4. Let N ≥ 1 N\geq1 N ≥ 1 α > 0 \alpha>0 α > 0 − min { 2 , N } < l < 0 -\min\{2,N\}<l<0 − min { 2 , N } < l < 0 C l C_l C l u 0 ≥ 0 u_0\geq0 u 0 ≥ 0 R N \mathbb{R}^N R N T max ( u 0 ) T_{\max}(u_0) T m a x ( u 0 ) u u u ( 0 , T max ( u 0 ) ) × R N (0,T_{\max}(u_0))\times\mathbb{R}^N ( 0 , T m a x ( u 0 )) × R N
(a) for all t ∈ ( 0 , T max ( u 0 ) ) t\in(0,T_{\max}(u_0)) t ∈ ( 0 , T m a x ( u 0 ))
2 2 + l α t 2 + l 2 [ ( e t Δ u 0 ) ( 0 ) ] α ≤ C l α . (36) \frac{2}{2+l}\alpha t^{\frac{2+l}{2}}\left[(e^{t\Delta}u_0)(0)\right]^\alpha\leq C_l^\alpha. \tag*{(36)} 2 + l 2 α t 2 2 + l [ ( e t Δ u 0 ) ( 0 ) ] α ≤ C l α . (36) (b) If 0 < T max ( u 0 ) < ∞ 0<T_{\max}(u_0)<\infty 0 < T m a x ( u 0 ) < ∞
2 2 + l α ( T max ( u 0 ) ) 2 + l 2 [ ( e T max ( u 0 ) Δ u 0 ) ( 0 ) ] α ≤ C l α . (37) \frac{2}{2+l}\alpha\left(T_{\max}(u_0)\right)^{\frac{2+l}{2}}\left[(e^{T_{\max}(u_0)\Delta}u_0)(0)\right]^\alpha\leq C_l^\alpha. \tag*{(37)} 2 + l 2 α ( T m a x ( u 0 ) ) 2 2 + l [ ( e T m a x ( u 0 ) Δ u 0 ) ( 0 ) ] α ≤ C l α . (37) (c) If 0 < T max ( u 0 ) < ∞ 0<T_{\max}(u_0)<\infty 0 < T m a x ( u 0 ) < ∞ u 0 = λ φ u_0=\lambda\varphi u 0 = λ φ λ > 0 \lambda>0 λ > 0
λ α 2 2 + l α ( T max ( λ φ ) ) 2 + l 2 [ ( e T max ( λ φ ) Δ φ ) ( 0 ) ] α ≤ C l α . (38) \lambda^\alpha\frac{2}{2+l}\alpha\left(T_{\max}(\lambda\varphi)\right)^{\frac{2+l}{2}}\left[(e^{T_{\max}(\lambda\varphi)\Delta}\varphi)(0)\right]^\alpha\leq C_l^\alpha. \tag*{(38)} λ α 2 + l 2 α ( T m a x ( λ φ ) ) 2 2 + l [ ( e T m a x ( λ φ ) Δ φ ) ( 0 ) ] α ≤ C l α . (38) Proof. (a) The proof is given in [6 ] and relies on inequality (2.14) together with the identity
( e ( τ − t ) Δ ∣ ⋅ ∣ − l α ) ( 0 ) = C l ( τ − t ) − l 2 α , \left(e^{(\tau-t)\Delta}|\cdot|^{-\frac{l}{\alpha}}\right)(0)=C_l(\tau-t)^{-\frac{l}{2\alpha}}, ( e ( τ − t ) Δ ∣ ⋅ ∣ − α l ) ( 0 ) = C l ( τ − t ) − 2 α l , where C l C_l C l [36 Lemma 4.1] .
(b) Inequality (2.18) is valid for all 0 < t < T max ( u 0 ) 0<t<T_{\max}(u_0) 0 < t < T m a x ( u 0 ) t = T max ( u 0 ) t=T_{\max}(u_0) t = T m a x ( u 0 )
(c) (2.20) follows directly from (2.19).
Remark 10. For − min { 2 , N } < l < 0 -\min\{2,N\}<l<0 − min { 2 , N } < l < 0 e t Δ ( ∣ ⋅ ∣ − l α ) ≤ e t Δ ( ∣ ⋅ ∣ − l α ) ( 0 ) e^{t\Delta}(|\cdot|^{-\frac{l}{\alpha}})\leq e^{t\Delta}(|\cdot|^{-\frac{l}{\alpha}})(0) e t Δ ( ∣ ⋅ ∣ − α l ) ≤ e t Δ ( ∣ ⋅ ∣ − α l ) ( 0 ) N = 1 N=1 N = 1 α = 1 4 \alpha=\frac{1}{4} α = 4 1 l = − 1 2 l=-\frac{1}{2} l = − 2 1 e t Δ ( ∣ ⋅ ∣ − l α ) ( x ) = x 2 + 2 t ≥ e t Δ ( ∣ ⋅ ∣ − l α ) ( 0 ) e^{t\Delta}(|\cdot|^{-\frac{l}{\alpha}})(x)=x^2+2t\geq e^{t\Delta}(|\cdot|^{-\frac{l}{\alpha}})(0) e t Δ ( ∣ ⋅ ∣ − α l ) ( x ) = x 2 + 2 t ≥ e t Δ ( ∣ ⋅ ∣ − α l ) ( 0 ) ( t , x ) ∈ ( 0 , ∞ ) × R (t,x)\in(0,\infty)\times\mathbb{R} ( t , x ) ∈ ( 0 , ∞ ) × R
Blow-up results
Let 0 < γ < N 0<\gamma<N 0 < γ < N X γ X_\gamma X γ [7 ] , see also [27 , 35 ] , by
X γ : = { ψ ∈ L l o c 1 ( R N ) ; ∣ ⋅ ∣ γ ψ ∈ L ∞ ( R N ) } = L γ ∞ , X_\gamma:=\left\{\psi\in L^1_{\mathrm{loc}}(\mathbb{R}^N);\ |\cdot|^\gamma\psi\in L^\infty(\mathbb{R}^N)\right\}=L^\infty_\gamma, X γ := { ψ ∈ L loc 1 ( R N ) ; ∣ ⋅ ∣ γ ψ ∈ L ∞ ( R N ) } = L γ ∞ , with norm ∥ ψ ∥ X γ : = ∥ ∣ ⋅ ∣ γ ψ ∥ L ∞ \|\psi\|_{X_\gamma}:=\||\cdot|^\gamma\psi\|_{L^\infty} ∥ ψ ∥ X γ := ∥∣ ⋅ ∣ γ ψ ∥ L ∞ ψ ∈ X γ \psi\in X_\gamma ψ ∈ X γ ∣ ψ ( x ) ∣ ≤ ∥ ψ ∥ X γ ∣ x ∣ − γ |\psi(x)|\leq\|\psi\|_{X_\gamma}|x|^{-\gamma} ∣ ψ ( x ) ∣ ≤ ∥ ψ ∥ X γ ∣ x ∣ − γ x ∈ R N x\in\mathbb{R}^N x ∈ R N
Y γ : = { ψ ∈ L l o c 1 ( R N ) ; ∣ ⋅ ∣ − γ ψ ∈ L 1 ( R N ) } . (39) Y_\gamma:=\left\{\psi\in L^1_{\mathrm{loc}}(\mathbb{R}^N);\ |\cdot|^{-\gamma}\psi\in L^1(\mathbb{R}^N)\right\}. \tag*{(39)} Y γ := { ψ ∈ L loc 1 ( R N ) ; ∣ ⋅ ∣ − γ ψ ∈ L 1 ( R N ) } . (39) It is straightforward to see that the predual of X γ \mathcal{X}_\gamma X γ Y γ \mathcal{Y}_\gamma Y γ K > 0 K>0 K > 0
B K : = { ψ ∈ X γ ; ∥ ψ ∥ X γ ≤ K } . (40) B_K:=\{\psi\in\mathcal{X}_\gamma;\quad\|\psi\|_{\mathcal{X}_\gamma}\le K\}. \tag*{(40)} B K := { ψ ∈ X γ ; ∥ ψ ∥ X γ ≤ K } . (40) Let B K ∗ B_K^* B K ∗ B K B_K B K ∗ * ∗ X γ \mathcal{X}_\gamma X γ B K ∗ B_K^* B K ∗ [5 ] ), and therefore complete and separable. Moreover, by [7 Proposition 2.1, p. 1110] , if ( ψ n ) n ≥ 1 (\psi_n)_{n\ge1} ( ψ n ) n ≥ 1 B K B_K B K ψ ∈ B K \psi\in B_K ψ ∈ B K ψ n → ψ \psi_n\to\psi ψ n → ψ B K ∗ B_K^* B K ∗ ψ n → ψ \psi_n\to\psi ψ n → ψ D ′ ( R N ) \mathcal{D}'(\mathbb{R}^N) D ′ ( R N ) ψ ∈ B K \psi\in B_K ψ ∈ B K [7 Formula (1.17), p. 1107] the following set
Z γ ( ψ ) : = { z ∈ B K ; ∃ λ n → ∞ such that λ n γ D λ n ψ → z in B K ∗ } , (41) \mathcal{Z}^\gamma(\psi):=\{z\in B_K;\quad\exists\lambda_n\to\infty\ \text{such that}\ \lambda_n^\gamma D_{\lambda_n}\psi\to z\ \text{in }B_K^*\}, \tag*{(41)} Z γ ( ψ ) := { z ∈ B K ; ∃ λ n → ∞ such that λ n γ D λ n ψ → z in B K ∗ } , (41) where D λ D_\lambda D λ (6) . By [7 Remark 2.4 (iii), p. 1112] , Z γ ( ψ ) \mathcal{Z}^\gamma(\psi) Z γ ( ψ ) B K ∗ B_K^* B K ∗
The proofs of Theorems 1.1 and 1.2, concerning blow-up for equation (1) , are entirely based on properties of the linear heat semigroup. The main ingredients are Propositions 2.2 and 2.4, which provide necessary conditions on the heat semigroup for the existence of nonnegative solutions on a given time interval, and [7 Proposition 3.8, p. 1123] . The following lemma will be needed. Although a proof can be found in [5 Exercise 4.26, p. 126] , we present a different one here.
Lemma 3.1. Let ( u n ) n ≥ 0 (u_n)_{n\ge0} ( u n ) n ≥ 0 L l o c 1 ( R N ) L^1_{\mathrm{loc}}(\mathbb{R}^N) L loc 1 ( R N )
(i) u n ≥ 0 u_n\ge0 u n ≥ 0 n ≥ 0 n\ge0 n ≥ 0
(ii) there exists u ∈ L l o c 1 ( R N ) u\in L^1_{\mathrm{loc}}(\mathbb{R}^N) u ∈ L loc 1 ( R N ) u n → u u_n\to u u n → u D ′ ( R N ) \mathcal{D}'(\mathbb{R}^N) D ′ ( R N )
Then u ≥ 0 u\ge0 u ≥ 0 R N \mathbb{R}^N R N
Proof. Since u n → u u_n\to u u n → u D ′ ( R N ) \mathcal{D}'(\mathbb{R}^N) D ′ ( R N )
∫ R N u n ( x ) φ ( x ) d x ⟶ ∫ R N u ( x ) φ ( x ) d x , for all φ ∈ C c ∞ ( R N ) . \int_{\mathbb{R}^N}u_n(x)\varphi(x)\,dx\longrightarrow\int_{\mathbb{R}^N}u(x)\varphi(x)\,dx,\quad\text{for all }\varphi\in C^\infty_c(\mathbb{R}^N). ∫ R N u n ( x ) φ ( x ) d x ⟶ ∫ R N u ( x ) φ ( x ) d x , for all φ ∈ C c ∞ ( R N ) . Using the fact that u n ≥ 0 u_n\ge0 u n ≥ 0 φ ∈ C c ∞ ( R N ) \varphi\in C^\infty_c(\mathbb{R}^N) φ ∈ C c ∞ ( R N ) φ ≥ 0 \varphi\ge0 φ ≥ 0
∫ R N u ( x ) φ ( x ) d x ≥ 0. (42) \int_{\mathbb{R}^N}u(x)\varphi(x)\,dx\ge0. \tag*{(42)} ∫ R N u ( x ) φ ( x ) d x ≥ 0. (42) In particular, (42) implies that
∫ K u ( x ) d x ≥ 0 , for every compact set K ⊂ R N . (43) \int_K u(x)\,dx\ge0,\quad\text{for every compact set }K\subset\mathbb{R}^N. \tag*{(43)} ∫ K u ( x ) d x ≥ 0 , for every compact set K ⊂ R N . (43) See the proof of [5 Corollary 4.24, p. 110] . Now, suppose by contradiction that A : = { x ∈ R N ; u ( x ) < 0 } A:=\{x\in\mathbb{R}^N;\ u(x)<0\} A := { x ∈ R N ; u ( x ) < 0 } [14 Proposition 7.5, p. 216] , there exists a compact set K 1 ⊂ A K_1\subset A K 1 ⊂ A
∫ K 1 u ( x ) d x < 0 , \int_{K_1}u(x)\,dx<0, ∫ K 1 u ( x ) d x < 0 , which contradicts (43) . □
Proof of Theorem 1.1. Let u 0 u_0 u 0 f f f T max ( f ) < ∞ T_{\max}(f)<\infty T m a x ( f ) < ∞ T max ( f ) = ∞ T_{\max}(f)=\infty T m a x ( f ) = ∞ (20) , we obtain
∥ e t Δ f ∥ L ∞ = ∥ D t e t Δ f ∥ L ∞ = ∥ e Δ D t f ∥ L ∞ , ∀ t > 0. \|e^{t\Delta}f\|_{L^\infty}=\|D_{\sqrt{t}}e^{t\Delta}f\|_{L^\infty}=\|e^{\Delta}D_{\sqrt{t}}f\|_{L^\infty},\quad\forall t>0. ∥ e t Δ f ∥ L ∞ = ∥ D t e t Δ f ∥ L ∞ = ∥ e Δ D t f ∥ L ∞ , ∀ t > 0. Thus,
t γ 2 ∥ e t Δ f ∥ L ∞ = ∥ e Δ ( t γ 2 D t f ) ∥ L ∞ , ∀ t > 0. t^{\frac{\gamma}{2}}\left\|e^{t\Delta}f\right\|_{L^\infty}=\left\|e^\Delta\left(t^{\frac{\gamma}{2}}D_{\sqrt{t}}f\right)\right\|_{L^\infty},\qquad\forall t>0. t 2 γ e t Δ f L ∞ = e Δ ( t 2 γ D t f ) L ∞ , ∀ t > 0. Let z ∈ L l o c 1 ( R N ) z\in L^1_{\mathrm{loc}}(\mathbb{R}^N) z ∈ L loc 1 ( R N ) z ≢ 0 z\not\equiv0 z ≡ 0 λ n → ∞ \lambda_n\to\infty λ n → ∞
λ n γ D λ n f → z in D ′ ( R N ) as n → ∞ . \lambda_n^\gamma D_{\lambda_n}f\to z\quad\text{in }D'(\mathbb{R}^N)\text{ as }n\to\infty. λ n γ D λ n f → z in D ′ ( R N ) as n → ∞. By [35 p. 169] , it follows that z ∈ Z γ ( f ) z\in Z^\gamma(f) z ∈ Z γ ( f ) B K ∗ B^*_{K} B K ∗ K K K t n : = λ n 2 → ∞ t_n:=\lambda_n^2\to\infty t n := λ n 2 → ∞ n → ∞ n\to\infty n → ∞
t n γ 2 D t n f → z in B K ∗ . t_n^{\frac{\gamma}{2}}D_{\sqrt{t_n}}f\to z\quad\text{in }B^*_{K}. t n 2 γ D t n f → z in B K ∗ . Applying [7 Proposition 3.8 (i), p. 1123] , we get
e Δ ( t n γ 2 D t n f ) → e Δ z in C 0 ( R N ) . e^\Delta\left(t_n^{\frac{\gamma}{2}}D_{\sqrt{t_n}}f\right)\to e^\Delta z\quad\text{in }C_0(\mathbb{R}^N). e Δ ( t n 2 γ D t n f ) → e Δ z in C 0 ( R N ) . Hence,
lim n → ∞ t n γ 2 ∥ e t n Δ f ∥ L ∞ = ∥ e Δ z ∥ L ∞ > 0. (44) \lim_{n\to\infty}t_n^{\frac{\gamma}{2}}\left\|e^{t_n\Delta}f\right\|_{L^\infty}=\left\|e^\Delta z\right\|_{L^\infty}>0. \tag*{(44)} n → ∞ lim t n 2 γ e t n Δ f L ∞ = e Δ z L ∞ > 0. (44) From (2.4), we obtain
t n γ 2 ∥ e t n Δ f ∥ L ∞ ≤ C l ( 2 + l 2 α ) 1 α t n γ 2 − 2 + l 2 α , ∀ n > 0. (45) t_n^{\frac{\gamma}{2}}\left\|e^{t_n\Delta}f\right\|_{L^\infty}\le C_l\left(\frac{2+l}{2\alpha}\right)^{\frac{1}{\alpha}}t_n^{\frac{\gamma}{2}-\frac{2+l}{2\alpha}},\qquad\forall n>0. \tag*{(45)} t n 2 γ e t n Δ f L ∞ ≤ C l ( 2 α 2 + l ) α 1 t n 2 γ − 2 α 2 + l , ∀ n > 0. (45) Passing to the limit as n → ∞ n\to\infty n → ∞ (45) and using (44) , we obtain a contradiction when α < 2 + l γ \alpha<\frac{2+l}{\gamma} α < γ 2 + l ∥ e Δ z ∥ L ∞ > 0 \left\|e^\Delta z\right\|_{L^\infty}>0 e Δ z L ∞ > 0 α = 2 + l γ \alpha=\frac{2+l}{\gamma} α = γ 2 + l n → ∞ n\to\infty n → ∞ ∥ e Δ z ∥ L ∞ > C l ( 2 + l 2 α ) 1 α \left\|e^\Delta z\right\|_{L^\infty}>C_l\left(\frac{2+l}{2\alpha}\right)^{\frac{1}{\alpha}} e Δ z L ∞ > C l ( 2 α 2 + l ) α 1 T max ( f ) < ∞ T_{\max}(f)<\infty T m a x ( f ) < ∞
Proof of Theorem 1.2. Let u 0 u_0 u 0 f f f T max ( f ) < ∞ T_{\max}(f)<\infty T m a x ( f ) < ∞ T max ( f ) = ∞ T_{\max}(f)=\infty T m a x ( f ) = ∞
t γ 2 ( e t Δ f ) ( 0 ) = t γ 2 ( D t e t Δ f ) ( 0 ) = t γ 2 ( e Δ D t f ) ( 0 ) , ∀ t > 0. t^{\frac{\gamma}{2}}\left(e^{t\Delta}f\right)(0)=t^{\frac{\gamma}{2}}\left(D_{\sqrt{t}}e^{t\Delta}f\right)(0)=t^{\frac{\gamma}{2}}\left(e^\Delta D_{\sqrt{t}}f\right)(0),\qquad\forall t>0. t 2 γ ( e t Δ f ) ( 0 ) = t 2 γ ( D t e t Δ f ) ( 0 ) = t 2 γ ( e Δ D t f ) ( 0 ) , ∀ t > 0. Thus, (2.18) implies that
t γ 2 ( D t e t Δ f ) ( 0 ) = t γ 2 ( e t Δ f ) ( 0 ) ≤ C l ( 2 + l 2 α ) 1 α t γ 2 − 2 + l 2 α , ∀ t > 0. (46) t^{\frac{\gamma}{2}}\left(D_{\sqrt{t}}e^{t\Delta}f\right)(0)=t^{\frac{\gamma}{2}}\left(e^{t\Delta}f\right)(0)\le C_l\left(\frac{2+l}{2\alpha}\right)^{\frac{1}{\alpha}}t^{\frac{\gamma}{2}-\frac{2+l}{2\alpha}},\qquad\forall t>0. \tag*{(46)} t 2 γ ( D t e t Δ f ) ( 0 ) = t 2 γ ( e t Δ f ) ( 0 ) ≤ C l ( 2 α 2 + l ) α 1 t 2 γ − 2 α 2 + l , ∀ t > 0. (46) Let z ∈ L l o c 1 ( R N ) z\in L^1_{\mathrm{loc}}(\mathbb{R}^N) z ∈ L loc 1 ( R N ) z ≢ 0 z\not\equiv0 z ≡ 0 λ n → ∞ \lambda_n\to\infty λ n → ∞
λ n γ D λ n f → z in D ′ ( R N ) as n → ∞ . \lambda_n^\gamma D_{\lambda_n}f\to z\quad\text{in }D'(\mathbb{R}^N)\text{ as }n\to\infty. λ n γ D λ n f → z in D ′ ( R N ) as n → ∞. Let t n : = λ n 2 t_n:=\lambda_n^2 t n := λ n 2 t n → ∞ t_n\to\infty t n → ∞ n → ∞ n\to\infty n → ∞
e Δ ( t n γ 2 D t n f ) → e Δ z in C 0 ( R N ) , e^\Delta\left(t_n^{\frac{\gamma}{2}}D_{\sqrt{t_n}}f\right)\to e^\Delta z\quad\text{in }C_0(\mathbb{R}^N), e Δ ( t n 2 γ D t n f ) → e Δ z in C 0 ( R N ) , and in particular,
lim n → ∞ t n γ 2 ( e Δ D t n f ) ( 0 ) = ( e Δ z ) ( 0 ) . \lim_{n\to\infty}t_n^{\frac{\gamma}{2}}\left(e^\Delta D_{\sqrt{t_n}}f\right)(0)=\left(e^\Delta z\right)(0). n → ∞ lim t n 2 γ ( e Δ D t n f ) ( 0 ) = ( e Δ z ) ( 0 ) . Using (1.12), Lemma 3.1 and the fact that z ≢ 0 z\not\equiv0 z ≡ 0
( e Δ z ) ( 0 ) > 0. (47) \left(e^\Delta z\right)(0)>0. \tag*{(47)} ( e Δ z ) ( 0 ) > 0. (47) Finally, replacing t t t t n t_n t n (46) and taking the limit n → ∞ n\to\infty n → ∞ α < 2 + l γ \alpha<\frac{2+l}{\gamma} α < γ 2 + l
( e Δ z ) ( 0 ) ≤ 0 , \left(e^\Delta z\right)(0)\le0, ( e Δ z ) ( 0 ) ≤ 0 , which contradicts (3.29). We also get a contradiction if α = 2 + l γ \alpha= \frac{2+l}{\gamma} α = γ 2 + l n → ∞ n \to\infty n → ∞
( e Δ z ) ( 0 ) > C l ( 2 + l 2 α ) 1 α . (e^{\Delta z})(0) > C_l\left(\frac{2+l}{2\alpha}\right)^{\frac{1}{\alpha}}. ( e Δ z ) ( 0 ) > C l ( 2 α 2 + l ) α 1 . Therefore, we conclude that T max ( f ) < ∞ T_{\max}(f) < \infty T m a x ( f ) < ∞ □ \square □
Upper bounds of the life-span
In this section, we exploit conditions (2.3), (2.7) and (2.18) for the existence of solutions. Conditions (2.3) and (2.18) imply that if u u u ( 0 , T max ( u 0 ) ) (0,T_{\max}(u_0)) ( 0 , T m a x ( u 0 ))
2 2 + l α t 2 + l 2 [ ( e t Δ u 0 ) ( 0 ) ] α ≤ C l α , (48) \frac{2}{2+l}\alpha t^{\frac{2+l}{2}}\left[(e^{t\Delta}u_0)(0)\right]^\alpha\leq C_l^\alpha, \tag*{(48)} 2 + l 2 α t 2 2 + l [ ( e t Δ u 0 ) ( 0 ) ] α ≤ C l α , (48) for all t ∈ ( 0 , T max ( u 0 ) ) t \in(0,T_{\max}(u_0)) t ∈ ( 0 , T m a x ( u 0 )) − min { 2 , N } < l < N α -\min\{2,N\}<l<N\alpha − min { 2 , N } < l < N α u 0 ≥ 0 u_0\geq0 u 0 ≥ 0 R N \mathbb{R}^N R N u 0 = λ φ u_0=\lambda\varphi u 0 = λ φ
λ α 2 2 + l α t 2 + l 2 [ ( e t Δ φ ) ( 0 ) ] α ≤ C l α , (49) \lambda^\alpha\frac{2}{2+l}\alpha t^{\frac{2+l}{2}}\left[(e^{t\Delta}\varphi)(0)\right]^\alpha\leq C_l^\alpha, \tag*{(49)} λ α 2 + l 2 α t 2 2 + l [ ( e t Δ φ ) ( 0 ) ] α ≤ C l α , (49) for all t ∈ ( 0 , T max ( λ φ ) ) t \in(0,T_{\max}(\lambda\varphi)) t ∈ ( 0 , T m a x ( λ φ )) φ ≥ 0 \varphi\geq0 φ ≥ 0 φ ≢ 0 \varphi\not\equiv0 φ ≡ 0 t > 0 t>0 t > 0 λ > C l ( 2 + l 2 α ) 1 α t − 2 + l 2 α [ ( e t Δ φ ) ( 0 ) ] − 1 \lambda>C_l\left(\frac{2+l}{2\alpha}\right)^{\frac{1}{\alpha}}t^{-\frac{2+l}{2\alpha}}\left[(e^{t\Delta}\varphi)(0)\right]^{-1} λ > C l ( 2 α 2 + l ) α 1 t − 2 α 2 + l [ ( e t Δ φ ) ( 0 ) ] − 1 t t t t t t
T max ( λ φ ) ≤ t . T_{\max}(\lambda\varphi)\leq t. T m a x ( λ φ ) ≤ t . Since t > 0 t>0 t > 0 t > 0 t>0 t > 0 A ( t ) > 0 A(t)>0 A ( t ) > 0 λ > A ( t ) \lambda>A(t) λ > A ( t ) T max ( λ φ ) ≤ t T_{\max}(\lambda\varphi)\leq t T m a x ( λ φ ) ≤ t φ \varphi φ
lim λ → ∞ T max ( λ φ ) = 0. (50) \lim_{\lambda\to\infty}T_{\max}(\lambda\varphi)=0. \tag*{(50)} λ → ∞ lim T m a x ( λ φ ) = 0. (50) We now give the proofs of the upper bounds.
Proof of Theorem 1.6. Let φ ∈ L l / α ∞ ( R N ) \varphi\in L^\infty_{l/\alpha}(\mathbb{R}^N) φ ∈ L l / α ∞ ( R N ) φ ≥ 0 \varphi\geq0 φ ≥ 0 φ ≢ 0 \varphi\not\equiv0 φ ≡ 0
λ α T max ( λ φ ) ≤ 1 α ( ∥ e Δ ∣ ⋅ ∣ − l / α ∥ L l / α ∞ ∥ e T max ( λ φ ) Δ φ ∥ L l / α ∞ ) α . \lambda^\alpha T_{\max}(\lambda\varphi)\leq\frac{1}{\alpha}\left(\frac{\|e^{\Delta}\,|\cdot|^{-l/\alpha}\|_{L^\infty_{l/\alpha}}}{\|e^{T_{\max}(\lambda\varphi)\Delta}\varphi\|_{L^\infty_{l/\alpha}}}\right)^\alpha. λ α T m a x ( λ φ ) ≤ α 1 ( ∥ e T m a x ( λ φ ) Δ φ ∥ L l / α ∞ ∥ e Δ ∣ ⋅ ∣ − l / α ∥ L l / α ∞ ) α . Consequently,
lim sup λ → ∞ λ α T max ( λ φ ) ≤ 1 α ( ∥ e Δ ∣ ⋅ ∣ − l / α ∥ L l / α ∞ lim inf λ → ∞ ∥ e T max ( λ φ ) Δ φ ∥ L l / α ∞ ) α . \limsup_{\lambda\to\infty}\lambda^\alpha T_{\max}(\lambda\varphi)\leq\frac{1}{\alpha}\left(\frac{\|e^{\Delta}\,|\cdot|^{-l/\alpha}\|_{L^\infty_{l/\alpha}}}{\liminf_{\lambda\to\infty}\|e^{T_{\max}(\lambda\varphi)\Delta}\varphi\|_{L^\infty_{l/\alpha}}}\right)^\alpha. λ → ∞ lim sup λ α T m a x ( λ φ ) ≤ α 1 ( lim inf λ → ∞ ∥ e T m a x ( λ φ ) Δ φ ∥ L l / α ∞ ∥ e Δ ∣ ⋅ ∣ − l / α ∥ L l / α ∞ ) α . Since T max ( λ φ ) → 0 T_{\max}(\lambda\varphi)\to0 T m a x ( λ φ ) → 0 λ → ∞ \lambda\to\infty λ → ∞ ∥ φ ∥ L l / α ∞ ≤ lim inf t → 0 ∥ e t Δ φ ∥ L l / α ∞ \|\varphi\|_{L^\infty_{l/\alpha}}\leq\liminf_{t\to0}\|e^{t\Delta}\varphi\|_{L^\infty_{l/\alpha}} ∥ φ ∥ L l / α ∞ ≤ lim inf t → 0 ∥ e t Δ φ ∥ L l / α ∞ ψ ∈ C ∞ ( R N ) \psi\in C^\infty(\mathbb{R}^N) ψ ∈ C ∞ ( R N )
(i) ∫ R N ( e t Δ φ ) ψ = ∫ R N φ ( e t Δ ψ ) \int_{\mathbb{R}^N}(e^{t\Delta}\varphi)\psi=\int_{\mathbb{R}^N}\varphi(e^{t\Delta}\psi) ∫ R N ( e t Δ φ ) ψ = ∫ R N φ ( e t Δ ψ )
∫ R N ( e t Δ φ ) ( x ) ψ ( x ) d x = ∫ R N ( ∫ R N G ( t , x − y ) φ ( y ) d y ) ψ ( x ) d x . \int_{\mathbb{R}^N}(e^{t\Delta}\varphi)(x)\psi(x)\,dx
=\int_{\mathbb{R}^N}\left(\int_{\mathbb{R}^N}G(t,x-y)\varphi(y)\,dy\right)\psi(x)\,dx. ∫ R N ( e t Δ φ ) ( x ) ψ ( x ) d x = ∫ R N ( ∫ R N G ( t , x − y ) φ ( y ) d y ) ψ ( x ) d x . We first verify that the double integral is absolutely convergent. In view of Remark 9, and using the assumptions 0 < l < N α 0<l<N\alpha 0 < l < N α ψ ∈ C c ∞ ( R N ) \psi\in C^\infty_c(\mathbb{R}^N) ψ ∈ C c ∞ ( R N )
∫ R N ∫ R N G ( t , x − y ) ∣ φ ( y ) ∣ ∣ ψ ( x ) ∣ d y d x ≤ ∫ R N ∫ R N G ( t , x − y ) ∥ φ ∥ L l / α ∞ ∣ y ∣ − l / α ∣ ψ ( x ) ∣ d y d x = ∥ φ ∥ L l / α ∞ ∫ R N ( e t Δ ⋅ ∣ ⋅ ∣ − l / α ) ( x ) ∣ ψ ( x ) ∣ d x ≤ ∥ φ ∥ L l / α ∞ ∫ R N ∣ e t Δ ⋅ ∣ ⋅ ∣ − l / α ∣ L l / α ∞ ∣ x ∣ − l / α ∣ ψ ( x ) ∣ d x ≤ C ∥ φ ∥ L l / α ∞ ∫ R N ∣ x ∣ − l / α ∣ ψ ( x ) ∣ d x < ∞ . \begin{aligned}
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G(t,x-y)|\varphi(y)||\psi(x)|\,dy\,dx
\leq\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}G(t,x-y)\|\varphi\|_{L^\infty_{l/\alpha}}|y|^{-l/\alpha}|\psi(x)|\,dy\,dx \\
&= \|\varphi\|_{L^\infty_{l/\alpha}}\int_{\mathbb{R}^N}\left(e^{t\Delta}\cdot|\cdot|^{-l/\alpha}\right)(x)|\psi(x)|\,dx \\
&\leq\|\varphi\|_{L^\infty_{l/\alpha}}\int_{\mathbb{R}^N}\left|e^{t\Delta}\cdot|\cdot|^{-l/\alpha}\right|_{L^\infty_{l/\alpha}}|x|^{-l/\alpha}|\psi(x)|\,dx \\
&\leq C\|\varphi\|_{L^\infty_{l/\alpha}}\int_{\mathbb{R}^N}|x|^{-l/\alpha}|\psi(x)|\,dx<\infty.
\end{aligned} ∫ R N ∫ R N G ( t , x − y ) ∣ φ ( y ) ∣∣ ψ ( x ) ∣ d y d x ≤ ∫ R N ∫ R N G ( t , x − y ) ∥ φ ∥ L l / α ∞ ∣ y ∣ − l / α ∣ ψ ( x ) ∣ d y d x = ∥ φ ∥ L l / α ∞ ∫ R N ( e t Δ ⋅ ∣ ⋅ ∣ − l / α ) ( x ) ∣ ψ ( x ) ∣ d x ≤ ∥ φ ∥ L l / α ∞ ∫ R N e t Δ ⋅ ∣ ⋅ ∣ − l / α L l / α ∞ ∣ x ∣ − l / α ∣ ψ ( x ) ∣ d x ≤ C ∥ φ ∥ L l / α ∞ ∫ R N ∣ x ∣ − l / α ∣ ψ ( x ) ∣ d x < ∞. Hence, Fubini’s theorem applies, and we can interchange the order of integration. Using the symmetry of the heat kernel, G ( t , x − y ) = G ( t , y − x ) G(t,x-y)=G(t,y-x) G ( t , x − y ) = G ( t , y − x )
∫ R N ( ∫ R N G ( t , x − y ) φ ( y ) d y ) ψ ( x ) d x = ∫ R N ( ∫ R N G ( t , x − y ) ψ ( x ) d x ) φ ( y ) d y = ∫ R N ( e t Δ ψ ) ( y ) φ ( y ) d y . \begin{aligned}
\int_{\mathbb{R}^N}\left(\int_{\mathbb{R}^N}G(t,x-y)\varphi(y)\,dy\right)\psi(x)\,dx
=
\int_{\mathbb{R}^N}\left(\int_{\mathbb{R}^N}G(t,x-y)\psi(x)\,dx\right)\varphi(y)\,dy \\
&=\int_{\mathbb{R}^N}(e^{t\Delta}\psi)(y)\varphi(y)\,dy.
\end{aligned} ∫ R N ( ∫ R N G ( t , x − y ) φ ( y ) d y ) ψ ( x ) d x = ∫ R N ( ∫ R N G ( t , x − y ) ψ ( x ) d x ) φ ( y ) d y = ∫ R N ( e t Δ ψ ) ( y ) φ ( y ) d y . This proves the desired identity.
(ii) ∫ R N φ ( e t Δ ψ ) → ∫ R N φ ψ \int_{\mathbb{R}^N}\varphi(e^{t\Delta}\psi)\to\int_{\mathbb{R}^N}\varphi\psi ∫ R N φ ( e t Δ ψ ) → ∫ R N φ ψ t → 0 t\to0 t → 0 ψ ∈ C c ∞ ( R N ) \psi\in C^\infty_c(\mathbb{R}^N) ψ ∈ C c ∞ ( R N )
∥ e t Δ ψ − ψ ∥ L ∞ ( R N ) → 0 , as t → 0. \|e^{t\Delta}\psi-\psi\|_{L^\infty(\mathbb{R}^N)}\to0,\qquad\text{as }t\to0. ∥ e t Δ ψ − ψ ∥ L ∞ ( R N ) → 0 , as t → 0. Consequently,
( e t Δ ψ ) ( x ) → ψ ( x ) , for every x ∈ R N . (e^{t\Delta}\psi)(x)\to\psi(x),\qquad\text{for every }x\in\mathbb{R}^N. ( e t Δ ψ ) ( x ) → ψ ( x ) , for every x ∈ R N . Hence,
φ ( x ) ( e t Δ ψ ) ( x ) → φ ( x ) ψ ( x ) , for every x ∈ R N . \varphi(x)(e^{t\Delta}\psi)(x)\to\varphi(x)\psi(x),\qquad\text{for every }x\in\mathbb{R}^N. φ ( x ) ( e t Δ ψ ) ( x ) → φ ( x ) ψ ( x ) , for every x ∈ R N . It remains to find an integrable function, independent of t t t ∣ φ ( x ) ( e t Δ ψ ) ( x ) ∣ |\varphi(x)(e^{t\Delta}\psi)(x)| ∣ φ ( x ) ( e t Δ ψ ) ( x ) ∣ supp ( ψ ) ⊂ B ( 0 , R ) = { y ∈ R N ; ∣ y ∣ ≤ R } \operatorname{supp}(\psi)\subset B(0,R)=\{y\in\mathbb{R}^N;\ |y|\leq R\} supp ( ψ ) ⊂ B ( 0 , R ) = { y ∈ R N ; ∣ y ∣ ≤ R } R > 0 R>0 R > 0 ∣ x ∣ ≥ 2 R |x|\geq2R ∣ x ∣ ≥ 2 R ∣ y ∣ ≤ R |y|\leq R ∣ y ∣ ≤ R ∣ x − y ∣ ≥ ∣ x ∣ − ∣ y ∣ ≥ ∣ x ∣ 2 |x-y|\geq|x|-|y|\geq\frac{|x|}{2} ∣ x − y ∣ ≥ ∣ x ∣ − ∣ y ∣ ≥ 2 ∣ x ∣
∣ ( e t Δ ψ ) ( x ) ∣ ≤ ( 4 π t ) − N 2 ∫ B ( 0 , R ) e − ∣ x − y ∣ 2 4 t ∣ ψ ( y ) ∣ d y ≤ C R t − N 2 e − ∣ x ∣ 2 16 t , ∀ ∣ x ∣ ≥ 2 R . (51) |(e^{t\Delta}\psi)(x)|
\leq(4\pi t)^{-\frac{N}{2}}\int_{B(0,R)}e^{-\frac{|x-y|^2}{4t}}|\psi(y)|\,dy
\leq CRt^{-\frac{N}{2}}e^{-\frac{|x|^2}{16t}},\qquad\forall|x|\geq2R. \tag*{(51)} ∣ ( e t Δ ψ ) ( x ) ∣ ≤ ( 4 π t ) − 2 N ∫ B ( 0 , R ) e − 4 t ∣ x − y ∣ 2 ∣ ψ ( y ) ∣ d y ≤ C R t − 2 N e − 16 t ∣ x ∣ 2 , ∀∣ x ∣ ≥ 2 R . (51) Moreover, for ∣ x ∣ ≥ 8 N |x|\geq\sqrt{8N} ∣ x ∣ ≥ 8 N t ↦ t − N 2 e − ∣ x ∣ 2 16 t t\mapsto t^{-\frac{N}{2}}e^{-\frac{|x|^2}{16t}} t ↦ t − 2 N e − 16 t ∣ x ∣ 2 ( 0 , 1 ] (0,1] ( 0 , 1 ]
t − N 2 e − ∣ x ∣ 2 16 t ≤ e − ∣ x ∣ 2 16 , ∀ 0 < t ≤ 1 , ∣ x ∣ ≥ 8 N . (52) t^{-\frac{N}{2}}e^{-\frac{|x|^2}{16t}}\leq e^{-\frac{|x|^2}{16}},\qquad\forall0<t\leq1,\ |x|\geq\sqrt{8N}. \tag*{(52)} t − 2 N e − 16 t ∣ x ∣ 2 ≤ e − 16 ∣ x ∣ 2 , ∀0 < t ≤ 1 , ∣ x ∣ ≥ 8 N . (52) Set M = max { 2 R , 8 N } M=\max\{2R,\sqrt{8N}\} M = max { 2 R , 8 N } (51) and (52) , we obtain, for 0 < t ≤ 1 0<t\leq1 0 < t ≤ 1 ∣ x ∣ ≥ M |x|\geq M ∣ x ∣ ≥ M
∣ φ ( x ) ( e t Δ ψ ) ( x ) ∣ ≤ C R ∥ φ ∥ L l / α ∞ ∣ x ∣ − l / α e − ∣ x ∣ 2 16 . |\varphi(x)(e^{t\Delta}\psi)(x)|\leq CR\|\varphi\|_{L^\infty_{l/\alpha}}|x|^{-l/\alpha}e^{-\frac{|x|^2}{16}}. ∣ φ ( x ) ( e t Δ ψ ) ( x ) ∣ ≤ C R ∥ φ ∥ L l / α ∞ ∣ x ∣ − l / α e − 16 ∣ x ∣ 2 . On the other hand, for ∣ x ∣ < M |x|<M ∣ x ∣ < M
∣ ( e t Δ ψ ) ( x ) ∣ ≤ ∥ ψ ∥ L ∞ , |(e^{t\Delta}\psi)(x)|\leq\|\psi\|_{L^\infty}, ∣ ( e t Δ ψ ) ( x ) ∣ ≤ ∥ ψ ∥ L ∞ , and consequently,
∣ φ ( x ) ( e t Δ ψ ) ( x ) ∣ ≤ ∥ φ ∥ L l / α ∞ ∥ ψ ∥ L ∞ ∣ x ∣ − l / α . |\varphi(x)(e^{t\Delta}\psi)(x)|\leq\|\varphi\|_{L^\infty_{l/\alpha}}\|\psi\|_{L^\infty}|x|^{-l/\alpha}. ∣ φ ( x ) ( e t Δ ψ ) ( x ) ∣ ≤ ∥ φ ∥ L l / α ∞ ∥ ψ ∥ L ∞ ∣ x ∣ − l / α . The dominated convergence theorem now proves the desired convergence.
Combining (i) and (ii), we conclude that e t Δ φ → φ e^{t\Delta}\varphi\to\varphi e t Δ φ → φ t → 0 t \to0 t → 0 D ′ ( R N ) D'(\mathbb{R}^{N}) D ′ ( R N ) [7 Proposition 2.1 (i), p. 1110] , e t Δ φ → φ e^{t\Delta}\varphi\to\varphi e t Δ φ → φ ∗ * ∗ t → 0 t \to0 t → 0 B max { 1 , C } ∥ φ ∥ L α ∞ ∗ B^{*}_{\max\{1,C\}\|\varphi\|_{L^\infty_\alpha}} B m a x { 1 , C } ∥ φ ∥ L α ∞ ∗ C C C B max { 1 , C } ∥ φ ∥ L α ∞ ∗ B^{*}_{\max\{1,C\}\|\varphi\|_{L^\infty_\alpha}} B m a x { 1 , C } ∥ φ ∥ L α ∞ ∗ [5 Proposition 3.13 (iii), p. 63] ,
∥ φ ∥ L α ∞ ≤ lim inf t → 0 ∥ e t Δ φ ∥ L α ∞ . \|\varphi\|_{L^\infty_\alpha} \le\liminf_{t\to0}\|e^{t\Delta}\varphi\|_{L^\infty_\alpha}. ∥ φ ∥ L α ∞ ≤ t → 0 lim inf ∥ e t Δ φ ∥ L α ∞ . □ \square □
Remark 11. It follows from [10 Lemma 2.1] that
∥ e t Δ φ ∥ L α ∞ ≤ C ∥ φ ∥ L α ∞ , ∀ t > 0. (53) \|e^{t\Delta}\varphi\|_{L^\infty_\alpha} \le C\|\varphi\|_{L^\infty_\alpha}, \qquad\forall t>0. \tag*{(53)} ∥ e t Δ φ ∥ L α ∞ ≤ C ∥ φ ∥ L α ∞ , ∀ t > 0. (53) Hence, lim sup t → 0 ∥ e t Δ φ ∥ L α ∞ ≤ C ∥ φ ∥ L α ∞ \limsup_{t\to0}\|e^{t\Delta}\varphi\|_{L^\infty_\alpha} \le C\|\varphi\|_{L^\infty_\alpha} lim sup t → 0 ∥ e t Δ φ ∥ L α ∞ ≤ C ∥ φ ∥ L α ∞ Theorem 1.6 yields ∥ φ ∥ L α ∞ ≤ lim inf t → 0 ∥ e t Δ φ ∥ L α ∞ \|\varphi\|_{L^\infty_\alpha} \le\liminf_{t\to0}\|e^{t\Delta}\varphi\|_{L^\infty_\alpha} ∥ φ ∥ L α ∞ ≤ lim inf t → 0 ∥ e t Δ φ ∥ L α ∞ ∥ e t Δ φ ∥ L α ∞ → ∥ φ ∥ L α ∞ \|e^{t\Delta}\varphi\|_{L^\infty_\alpha} \to\|\varphi\|_{L^\infty_\alpha} ∥ e t Δ φ ∥ L α ∞ → ∥ φ ∥ L α ∞ t → 0 t \to0 t → 0 [34 Proof of Theorem 1.6, p. 595] does not apply directly.
Proof of Theorem 1.7 . By inequality (38) , we have
λ 2 α 2 + ℓ T max ( λ φ ) ≤ ( 2 + ℓ 2 α ) 2 2 + ℓ [ C ℓ ( e T max ( λ φ ) Δ φ ) ( 0 ) ] 2 α 2 + ℓ . (54) \lambda^{\frac{2\alpha}{2+\ell}}T_{\max}(\lambda\varphi) \le\left(\frac{2+\ell}{2\alpha}\right)^{\frac{2}{2+\ell}}\left[\frac{C_\ell}{\left(e^{T_{\max}(\lambda\varphi)\Delta}\varphi\right)(0)}\right]^{\frac{2\alpha}{2+\ell}}. \tag*{(54)} λ 2 + ℓ 2 α T m a x ( λ φ ) ≤ ( 2 α 2 + ℓ ) 2 + ℓ 2 [ ( e T m a x ( λ φ ) Δ φ ) ( 0 ) C ℓ ] 2 + ℓ 2 α . (54) Passing to the lim sup as λ → ∞ \lambda\to\infty λ → ∞ (54) , and using (50) together with the fact that e t Δ e^{t\Delta} e t Δ C 0 C_0 C 0 C 0 ( R N ) C_0(\mathbb{R}^{N}) C 0 ( R N ) □ \square □
Proof of Theorem 1.8 . Let
φ ~ ( x ) = { ω ( x ) ∣ x ∣ − γ , ∣ x ∣ ≤ ε , 0 , ∣ x ∣ > ε . (55) \widetilde{\varphi}(x)=
\begin{cases}
\omega(x)|x|^{-\gamma}, & |x|\le\varepsilon,\\
0, & |x|>\varepsilon.
\end{cases} \tag*{(55)} φ ( x ) = { ω ( x ) ∣ x ∣ − γ , 0 , ∣ x ∣ ≤ ε , ∣ x ∣ > ε . (55) Since 0 ≤ φ ~ ≤ φ 0\le\widetilde{\varphi}\le\varphi 0 ≤ φ ≤ φ T max ( λ φ ) ≤ T max ( λ φ ~ ) T_{\max}(\lambda\varphi)\le T_{\max}(\lambda\widetilde{\varphi}) T m a x ( λ φ ) ≤ T m a x ( λ φ ) T max ( λ φ ~ ) T_{\max}(\lambda\widetilde{\varphi}) T m a x ( λ φ ) [34 p. 595] , We have
∥ e t Δ φ ~ ∥ ∞ = ∥ D t e t Δ φ ~ ∥ ∞ = ∥ e Δ D t φ ~ ∥ ∞ = t − γ 2 ∥ e Δ [ t γ 2 D t φ ~ ] ∥ ∞ . (56) \|e^{t\Delta}\widetilde{\varphi}\|_\infty=\|D_{\sqrt{t}}e^{t\Delta}\widetilde{\varphi}\|_\infty=\|e^\Delta D_{\sqrt{t}}\widetilde{\varphi}\|_\infty=t^{-\frac{\gamma}{2}}\|e^\Delta[t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}]\|_\infty. \tag*{(56)} ∥ e t Δ φ ∥ ∞ = ∥ D t e t Δ φ ∥ ∞ = ∥ e Δ D t φ ∥ ∞ = t − 2 γ ∥ e Δ [ t 2 γ D t φ ] ∥ ∞ . (56) Since t γ 2 D t φ ~ → ω ∣ ⋅ ∣ − γ t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\to\omega|\cdot|^{-\gamma} t 2 γ D t φ → ω ∣ ⋅ ∣ − γ D ′ ( R N ) D'(\mathbb{R}^{N}) D ′ ( R N ) t → 0 t\to0 t → 0 [7 Proposition 3.8 (i), page 1123] , that
t γ 2 ∥ e t Δ φ ~ ∥ ∞ → ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ , as t → 0. t^{\frac{\gamma}{2}}\|e^{t\Delta}\widetilde{\varphi}\|_\infty\to\|e^\Delta(\omega|\cdot|^{-\gamma})\|_{L^\infty}, \qquad\text{as }t\to0. t 2 γ ∥ e t Δ φ ∥ ∞ → ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ , as t → 0. Since T max ( λ φ ) → 0 T_{\max}(\lambda\varphi)\to0 T m a x ( λ φ ) → 0 λ → ∞ \lambda\to\infty λ → ∞ (50) , we deduce from (24) that
lim sup λ → ∞ λ 2 α 2 + ℓ ( T max ( λ φ ~ ) ) 1 − 2 + ℓ 2 α ≤ ( 2 + ℓ 2 α ) 2 2 + ℓ [ C ℓ ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ ] 2 α 2 + ℓ , \limsup_{\lambda\to\infty}\lambda^{\frac{2\alpha}{2+\ell}}\left(T_{\max}(\lambda\widetilde{\varphi})\right)^{1-\frac{2+\ell}{2\alpha}}\le\left(\frac{2+\ell}{2\alpha}\right)^{\frac{2}{2+\ell}}\left[\frac{C_\ell}{\|e^\Delta(\omega|\cdot|^{-\gamma})\|_{L^\infty}}\right]^{\frac{2\alpha}{2+\ell}}, λ → ∞ lim sup λ 2 + ℓ 2 α ( T m a x ( λ φ ) ) 1 − 2 α 2 + ℓ ≤ ( 2 α 2 + ℓ ) 2 + ℓ 2 [ ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ C ℓ ] 2 + ℓ 2 α , which completes the proof. □ \square □
Proof of Theorem 1.9 . It suffices to estimate T max ( λ φ ~ ) T_{\max}(\lambda\widetilde{\varphi}) T m a x ( λ φ ) φ ~ \widetilde{\varphi} φ (55) . For all t > 0 t>0 t > 0
t γ 2 ( e t Δ φ ~ ) ( 0 ) = [ e Δ ( t γ 2 D t φ ~ ) ] ( 0 ) . (57) t^{\frac{\gamma}{2}}\left(e^{t\Delta}\widetilde{\varphi}\right)(0)=\left[e^\Delta\left(t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\right)\right](0). \tag*{(57)} t 2 γ ( e t Δ φ ) ( 0 ) = [ e Δ ( t 2 γ D t φ ) ] ( 0 ) . (57) Since t γ 2 D t φ ~ → ω ∣ ⋅ ∣ − γ t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\to\omega|\cdot|^{-\gamma} t 2 γ D t φ → ω ∣ ⋅ ∣ − γ D ′ ( R N ) D'(\mathbb{R}^{N}) D ′ ( R N ) t → 0 t\to0 t → 0 [7 Proposition 3.8 (i), p. 1123] , that
e Δ ( t γ 2 D t φ ~ ) → e Δ ( ω ∣ ⋅ ∣ − γ ) in C 0 ( R N ) , as t → 0 , e^\Delta\left(t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\right)\to e^\Delta(\omega|\cdot|^{-\gamma})\quad\text{in }C_0(\mathbb{R}^{N}),\quad\text{as }t\to0, e Δ ( t 2 γ D t φ ) → e Δ ( ω ∣ ⋅ ∣ − γ ) in C 0 ( R N ) , as t → 0 , and consequently,
t γ 2 ( e t Δ ~ φ ~ ) ( 0 ) → e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) , as t → 0. (58) t^{\frac{\gamma}{2}}\left(e^{t\widetilde{\Delta}}\widetilde{\varphi}\right)(0)\to e^{\Delta}\left(\omega|\cdot|^{-\gamma}\right)(0),\qquad\text{as }t\to0. \tag*{(58)} t 2 γ ( e t Δ φ ) ( 0 ) → e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) , as t → 0. (58) Finally, since T max ( λ φ ) → 0 T_{\max}(\lambda\varphi)\to0 T m a x ( λ φ ) → 0 λ → ∞ \lambda\to\infty λ → ∞ (38) with (58) yields
lim sup λ → ∞ λ 2 α 2 + l ( T max ( λ φ ) ) 1 − 2 α γ 2 + l ≤ ( 2 + l 2 α ) 2 2 + l [ C l e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) ] 2 α 2 + l . \limsup_{\lambda\to\infty}\lambda^{\frac{2\alpha}{2+l}}\left(T_{\max}(\lambda\varphi)\right)^{1-\frac{2\alpha\gamma}{2+l}}\le\left(\frac{2+l}{2\alpha}\right)^{\frac{2}{2+l}}\left[\frac{C_l}{e^{\Delta}(\omega|\cdot|^{-\gamma})(0)}\right]^{\frac{2\alpha}{2+l}}. λ → ∞ lim sup λ 2 + l 2 α ( T m a x ( λ φ ) ) 1 − 2 + l 2 α γ ≤ ( 2 α 2 + l ) 2 + l 2 [ e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) C l ] 2 + l 2 α . which concludes the proof. □
Proof of Theorem 1.10. Let
φ ~ ( x ) = { 0 , ∣ x ∣ < R , ω ( x ) ∣ x ∣ − γ , ∣ x ∣ ≥ R . (59) \widetilde{\varphi}(x)=
\begin{cases}
0, & |x|<R,\\
\omega(x)|x|^{-\gamma}, & |x|\ge R.
\end{cases} \tag*{(59)} φ ( x ) = { 0 , ω ( x ) ∣ x ∣ − γ , ∣ x ∣ < R , ∣ x ∣ ≥ R . (59) Since φ ≥ φ ~ \varphi\ge\widetilde{\varphi} φ ≥ φ T max ( λ φ ~ ) T_{\max}(\lambda\widetilde{\varphi}) T m a x ( λ φ ) [34 p. 596] , the calculation in (56) gives
∥ e t Δ φ ~ ∥ L ∞ = t − γ 2 ∥ e Δ [ t γ 2 D t φ ~ ] ∥ L ∞ . \left\|e^{t\Delta}\widetilde{\varphi}\right\|_{L^\infty}=t^{-\frac{\gamma}{2}}\left\|e^{\Delta}\left[t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\right]\right\|_{L^\infty}. e t Δ φ L ∞ = t − 2 γ e Δ [ t 2 γ D t φ ] L ∞ . Moreover, t γ 2 D t φ ~ → ω ∣ ⋅ ∣ − γ t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\to\omega|\cdot|^{-\gamma} t 2 γ D t φ → ω ∣ ⋅ ∣ − γ t → ∞ t\to\infty t → ∞ D ′ ( R N ) \mathcal{D}'(\mathbb{R}^{N}) D ′ ( R N ) [7 Proposition 3.8 (i), page 1123] that
t γ 2 ∥ e t Δ φ ~ ∥ L ∞ = ∥ e Δ [ t γ 2 D t φ ~ ] ∥ L ∞ → ∥ e Δ [ ω ∣ ⋅ ∣ − γ ] ∥ L ∞ , as t → ∞ . (60) t^{\frac{\gamma}{2}}\left\|e^{t\Delta}\widetilde{\varphi}\right\|_{L^\infty}=\left\|e^{\Delta}\left[t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\right]\right\|_{L^\infty}\to\left\|e^{\Delta}\left[\omega|\cdot|^{-\gamma}\right]\right\|_{L^\infty},\qquad\text{as }t\to\infty. \tag*{(60)} t 2 γ e t Δ φ L ∞ = e Δ [ t 2 γ D t φ ] L ∞ → e Δ [ ω ∣ ⋅ ∣ − γ ] L ∞ , as t → ∞. (60) If 0 < t < T max ( λ φ ~ ) 0<t<T_{\max}(\lambda\widetilde{\varphi}) 0 < t < T m a x ( λ φ ) (22) we must have
2 2 + l α λ α t 2 + l 2 α − γ 2 [ t γ 2 ∥ e t Δ φ ~ ∥ L ∞ ] α ≤ C l α . (61) \frac{2}{2+l}\alpha\lambda^\alpha t^{\frac{2+l}{2\alpha}-\frac{\gamma}{2}}\left[t^{\frac{\gamma}{2}}\left\|e^{t\Delta}\widetilde{\varphi}\right\|_{L^\infty}\right]^\alpha\le C_l^\alpha. \tag*{(61)} 2 + l 2 α λ α t 2 α 2 + l − 2 γ [ t 2 γ e t Δ φ L ∞ ] α ≤ C l α . (61) It follows that if γ < 2 + l α \gamma<\frac{2+l}{\alpha} γ < α 2 + l T max ( λ φ ~ ) < ∞ T_{\max}(\lambda\widetilde{\varphi})<\infty T m a x ( λ φ ) < ∞ λ > 0 \lambda>0 λ > 0 α ≤ 2 + l N \alpha\le\frac{2+l}{N} α ≤ N 2 + l α > 2 + l N \alpha>\frac{2+l}{N} α > N 2 + l Theorem 1.1 and Remark 2 (iv). On the other hand, it follows from Theorem 1.5 when l α < γ < min { N , 2 + l α } \frac{l}{\alpha}<\gamma<\min\{N,\frac{2+l}{\alpha}\} α l < γ < min { N , α 2 + l } Theorem 1.4 when γ = l α \gamma=\frac{l}{\alpha} γ = α l T max ( λ φ ~ ) → ∞ T_{\max}(\lambda\widetilde{\varphi})\to\infty T m a x ( λ φ ) → ∞ λ → 0 \lambda\to0 λ → 0 T max ( λ φ ~ ) → ∞ T_{\max}(\lambda\widetilde{\varphi})\to\infty T m a x ( λ φ ) → ∞ λ → 0 \lambda\to0 λ → 0 t = T max ( λ φ ~ ) t=T_{\max}(\lambda\widetilde{\varphi}) t = T m a x ( λ φ ) (61) and letting λ → 0 \lambda\to0 λ → 0 (60) , we obtain
lim sup λ → 0 λ ( 2 + l 2 α − γ 2 ) − 1 T max ( λ φ ~ ) ≤ [ C l ( 2 α / ( 2 + l ) ) 1 / α ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ ] ( 2 + l 2 α − γ 2 ) − 1 . \limsup_{\lambda\to0}\lambda^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}T_{\max}(\lambda\widetilde{\varphi})\le\left[\frac{C_l}{(2\alpha/(2+l))^{1/\alpha}\left\|e^{\Delta}(\omega|\cdot|^{-\gamma})\right\|_{L^\infty}}\right]^{\left(\frac{2+l}{2\alpha}-\frac{\gamma}{2}\right)^{-1}}. λ → 0 lim sup λ ( 2 α 2 + l − 2 γ ) − 1 T m a x ( λ φ ) ≤ [ ( 2 α / ( 2 + l ) ) 1/ α ∥ e Δ ( ω ∣ ⋅ ∣ − γ ) ∥ L ∞ C l ] ( 2 α 2 + l − 2 γ ) − 1 . This completes the proof. □
Proof of Theorem 1.11. Let φ ~ \widetilde{\varphi} φ (59) . It is sufficient to derive an upper bound for T max ( λ φ ~ ) T_{\max}(\lambda\widetilde{\varphi}) T m a x ( λ φ ) γ < 2 + l α \gamma<\frac{2+l}{\alpha} γ < α 2 + l T max ( λ φ ~ ) < ∞ T_{\max}(\lambda\widetilde{\varphi})<\infty T m a x ( λ φ ) < ∞ λ > 0 \lambda>0 λ > 0 Theorem 1.2 , and Remark 2 (iv). Next, replacing φ ~ \widetilde{\varphi} φ φ ~ \widetilde{\varphi} φ (57) , we obtain
t γ 2 ( e t Δ φ ~ ) ( 0 ) = [ e Δ ( t γ 2 D t φ ~ ) ] ( 0 ) . t^{\frac{\gamma}{2}}\left(e^{t\Delta}\widetilde{\varphi}\right)(0)=\left[e^{\Delta}\left(t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\right)\right](0). t 2 γ ( e t Δ φ ) ( 0 ) = [ e Δ ( t 2 γ D t φ ) ] ( 0 ) . Moreover, as t → ∞ t\to\infty t → ∞ t γ 2 D t φ ~ → ω ∣ ⋅ ∣ − γ t^{\frac{\gamma}{2}}D_{\sqrt{t}}\widetilde{\varphi}\to\omega|\cdot|^{-\gamma} t 2 γ D t φ → ω ∣ ⋅ ∣ − γ D ′ ( R N ) \mathcal{D}'(\mathbb{R}^{N}) D ′ ( R N ) [7 Proposition 3.8, p. 1123] , it follows that
t γ 2 ( e t Δ φ ~ ) ( 0 ) → ( e Δ ( ω ∣ ⋅ ∣ − γ ) ) ( 0 ) , as t → ∞ . t^{\frac{\gamma}{2}}\left(e^{t\Delta}\widetilde{\varphi}\right)(0)\to\left(e^{\Delta}(\omega|\cdot|^{-\gamma})\right)(0),\qquad\text{as }t\to\infty. t 2 γ ( e t Δ φ ) ( 0 ) → ( e Δ ( ω ∣ ⋅ ∣ − γ ) ) ( 0 ) , as t → ∞. It follows from Theorem 1.5 that T max ( λ φ ~ ) → ∞ T_{\max}(\lambda\widetilde{\varphi})\to\infty T m a x ( λ φ ) → ∞ λ → 0 \lambda\to0 λ → 0 (38) , implies that
lim sup λ → 0 λ 2 α 2 + l ( T max ( λ φ ~ ) ) 1 − 2 α γ 2 + l ≤ ( 2 + l 2 α ) 2 2 + l [ C l e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) ] 2 α 2 + l . \limsup_{\lambda\to0}\lambda^{\frac{2\alpha}{2+l}}\left(T_{\max}(\lambda\widetilde{\varphi})\right)^{1-\frac{2\alpha\gamma}{2+l}}\le\left(\frac{2+l}{2\alpha}\right)^{\frac{2}{2+l}}\left[\frac{C_l}{e^{\Delta}(\omega|\cdot|^{-\gamma})(0)}\right]^{\frac{2\alpha}{2+l}}. λ → 0 lim sup λ 2 + l 2 α ( T m a x ( λ φ ) ) 1 − 2 + l 2 α γ ≤ ( 2 α 2 + l ) 2 + l 2 [ e Δ ( ω ∣ ⋅ ∣ − γ ) ( 0 ) C l ] 2 + l 2 α . References
Byrame Ben Slimene, Taibah University, College of Science, Department of Mathematics, 42353 AL-Madinah, Saudi Arabia, e-mail: byramebenslimene@yahoo.fr
Slim Tayachi, Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, Laboratoire Équations aux Dérivées Partielles LR03ES04, 2092 Tunis, Tunisia, e-mail: slim.tayachi@fst.rnu.tn
