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

We investigate forward and backward smoothing effects in Lebesgue spaces $L^p$ and ${\mathcal{M}}^p:=L^{p,\mathrm{\infty }}$ for the Cauchy problem associated to the nonlinear and nonlocal fractional diffusion equation ${\mathrm{\partial }}_{t}u+(-\mathrm{\Delta }{)}^{\frac{\sigma }{2}}\mathrm{\mid }u{\mathrm{\mid }}^{m-1}u=0$ in ${\mathbb{R}}^N$, $0<\sigma <2$, in the very fast range $0<m\le m_c:=\frac{N-\sigma }{N}$. We prove that very weak solutions have an ${\mathcal{M}}^p$–$L^{\mathrm{\infty }}$ smoothing effect if $p>p^{\ast }:=\frac{N}{\sigma }(1-m)$, and we construct counterexamples showing the failure of any $L^p$–${\mathcal{M}}^q$ forward ($q>p$) smoothing effect if $1<p<p^{\ast }$, $m<m_c$ or $L^1$–${\mathcal{M}}^q$ ($q>1$) if $m=m_c$. We also prove a backward ${\mathcal{M}}^p$–$L^1$ smoothing effect whenever $1\le p<p^{\ast }$, $m<m_c$, and we construct counterexamples showing that there is no $L^p$–${\mathcal{M}}^q$ backward ($q<p$) smoothing effect if $p>p^{\ast }$. Regarding the threshold value $p=p^{\ast }$, we prove that all solutions starting in ${\mathcal{M}}^{p^{\ast }}$ become extinct in finite time, and show the failure of any ${\mathcal{M}}^{p^{\ast }}$–${\mathcal{M}}^q$ smoothing before extinction for any $q\mathrm{\ne }{p}^{\ast }$. The construction of counterexamples is based on new uniqueness and comparison results for very weak solutions, combined with the existence of self-similar solutions with suitable properties. The same approach yields new counterexamples for both forward and backward smoothing effects also in the local case $\sigma =2$.