Fractional very fast diffusion equations in Lebesgue spaces: uniqueness and smoothing effects

Introduction

We give sharp conditions for the existence of smoothing effects for the nonlinear fractional Cauchy problem

in the so-called very fast range, $m \in(0,m_c]$, with $m_c = \frac{N-\sigma}{N} > 0$. Observe that necessarily $N > \sigma$. We assume a fractional order $\sigma\in(0,2)$ for the fractional Laplacian $(-\Delta)^{\frac{\sigma}{2}}$.

We say that there is an X–Y smoothing effect if all solutions with initial datum in X belong to X $\cap$ Y for almost every positive time. We will focus here initially on $L^p$–$L^q$ smoothing effects, $p \in[1,\infty)$, $p < q \le\infty$. However, it turns out that the spaces that mark the threshold between different behaviours are the weak Lebesgue spaces $L^{p,\infty}(\mathbb{R}^N)$, $p > 1$, slightly larger than $L^p(\mathbb{R}^N)$, also known as Marcinkiewicz spaces, denoted here by $\mathcal{M}^p(\mathbb{R}^N)$. Therefore, we will also perform our analysis within them. In order to simplify the reading and without loss of generality we restrict to nonnegative solutions.

2020 Mathematics Subject Classification. 35R11, 35B51, 35A02, 35B65.

Key words and phrases. Nonlinear fractional diffusion, uniqueness, comparison, very weak solutions, regularization.

The name very fast diffusion in the title comes from applications in the local case $\sigma=2$, i.e., $\partial_t u+(-\Delta)u^m=0$, which corresponds to slow diffusion if $m>1$, fast diffusion if $m\in(m_c,1)$, and very fast diffusion if $m\in(0,m_c)$. In this local case the critical exponent is $m_c:=\frac{(N-2)_+}{N}$. The first proof of a smoothing effect ($L^1$–$L^\infty$ if $m>m_c$) for local diffusion was given in [1, 20]. Later, in [17], Vázquez proved an $\mathcal{M}^p$–$L^\infty$ smoothing effect if $m<m_c$ under the condition $p>\frac{N}{2}(1-m)$.

As for the nonlocal problem (1.1), existence and uniqueness of a weak solution was proved in [6] both if $m>m_c$ and $u_0\in L^1(\mathbb{R}^N)$, or if $m\in(0,m_c]$ and $u_0\in L^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$, $p>p^*:=\frac{N}{\sigma}(1-m)$. A crucial point of the existence proof is the fact that solutions immediately enter into $L^\infty(\mathbb{R}^N)$, while staying in the same space as the initial datum: if $m\in(0,m_c]$ and $p>p^*$ there is an $(L^1\cap L^p)$–$L^\infty$ smoothing effect.

The exponent $p^*$ arises naturally from scaling: problem (1.1) is invariant under the mapping $(x,t,u)\mapsto(\lambda^\beta x,\lambda t,\lambda^\alpha u)$ provided $\alpha(m-1)+\beta\sigma=1$, and for the datum $|x|^{-\frac{N}{p}}$ to be preserved one needs additionally $\alpha=\frac{N\beta}{p}$; this system has a unique solution $(\alpha,\beta)$ if and only if $p\ne p^*$. When $p\ne p^*$, this gives rise to self-similar solutions of the form $t^{-\alpha}S(t^{-\beta}|x|)$, which will play a key role in our analysis. The critical case $p=p^*$, where this scaling argument breaks down, corresponds instead to solutions in separated variables that become extinct in finite time; see Theorem 4.2. Observe also that $p^*=1$ in the borderline case $m=m_c$.

Forward smoothing effects. Two natural questions arise.

Is there an $\mathcal{M}^p$–$L^\infty$ smoothing effect if $p>p^*$, without requiring $u_0\in L^1(\mathbb{R}^N)$?

Notice that a positive answer would imply an $L^p$–$L^\infty$ smoothing effect, since $L^p(\mathbb{R}^N)\subset\mathcal{M}^p(\mathbb{R}^N)$.

Under this weaker assumption on the initial datum, the existence of a solution, in some suitable sense, has to be justified. The existence of very weak solutions with large initial data, for all $m>0$, was addressed in [3]: if the initial datum belongs to the weighted space $L^1_v(\mathbb{R}^N)=L^1(\mathbb{R}^N;v\,dx)$ with $v$ in a suitable class, there is a very weak solution that moreover stays in $L^1_v(\mathbb{R})$ for any later time. Uniqueness for such solutions when $m>m_c$ was proved in [13]. Uniqueness for $m\in(0,m_c]$ remained an open problem, and will be proved here; see Theorem 3.3. Beyond its intrinsic interest, it has several consequences that will be essential in our analysis of smoothing effects.

If the initial datum belongs to $L^p(\mathbb{R}^N)$ or $\mathcal{M}^p(\mathbb{R}^N)$, it belongs to $L^1_v(\mathbb{R}^N)$ for some weights $v$ in the admissible class. Hence, there is a very weak solution to the problem. Moreover, we will prove that this solution stays in the same space as the initial datum, $L^p(\mathbb{R}^N)$ or $\mathcal{M}^p(\mathbb{R}^N)$, for later times, thus showing the existence of a very weak $L^p$ or $\mathcal{M}^p$ solution in the sense of Definition 2.2; see Theorems 4.1 and 4.5. For the rest of the paper we will work in these classes of solutions.

A partial answer to our first question was given in [19], where the authors prove the desired $\mathcal{M}^p$–$L^\infty$ smoothing effect, but only for minimal very weak solutions in $L^1_v(\mathbb{R}^N)$ (remember that uniqueness was not available). Thus, the possibility of larger solutions staying unbounded was still there. Our uniqueness result rules out this possibility, since there is no other solution different from the minimal one, and we obtain a positive answer to our question; see Theorem 4.6.

We turn now our attention to the second question.

Is there any $L^p$ (or $\mathcal{M}^p$)–$L^q$ (or $\mathcal{M}^q$) smoothing effect with $q>p$, if $p\le p^*$?

The answer is no. This is the content of Theorem 6.1(i). If $p < p^*$ we construct counterexamples in self-similar form. The same approach can also be used to give counterexamples in the local case, that were not available up to now for $q \in(p,p^*]$. The proof when $p = p^*$ follows an idea from [4] for the local case, based on a comparison principle. Such comparison in our nonlocal setting was not known in the range $m \in(0,m_{\mathrm{c}}]$; it will follow as a corollary of our uniqueness result; see Theorem 3.4.

Backward smoothing effects

Up to now we have only considered the possibility of passing from an $L^p$ (or $\mathcal{M}^p$) space to an $L^q$ (or $\mathcal{M}^q$) space with $q > p$. These are the so-called forward smoothing effects. But we could think about the possibility of entering into integrability spaces with $q < p$ (notice that this question made no sense for solutions living in $L^1(\mathbb{R}^N) \cap L^p(\mathbb{R}^N)$ as the ones considered in [6]). Such a phenomenon is named as a backward smoothing effect. New questions arise.

Is there any $L^p$ (or $\mathcal{M}^p$)–$L^q$ (or $\mathcal{M}^q$), backward ($q < p$) smoothing effect?

Though there is no forward smoothing effect for initial data in $\mathcal{M}^p$ when $p < p^*$, we will show in Theorem 5.1 that there is a backward $\mathcal{M}^p$–$L^1$ smoothing effect: $\mathcal{M}^p$ solutions will become instantaneously integrable. The corresponding analysis for the local case was developed in [17]. One may then wonder whether such backward smoothing effects could happen if $p \geq p^*$. At this respect we use the self-similar solutions constructed in [19] to show that this is not the case; see Theorem 6.1(ii). A similar approach can be used to give counterexamples for the local case, that were not available up to now.

The following table summarizes our results on the forward and backward $L^p$–$\mathcal{M}^q$ and $\mathcal{M}^p$–$L^q$ smoothing effects.