Sharp extinction rates for positive solutions of fast diffusion equations

Abstract

Let $s\in (0,1]$ and $N>2s$. It is known that positive solutions to the (fractional) fast diffusion equation ${\partial }_{t}u+(-\hat{I}”{)}^s(u^{\frac{N-2s}{N+2s}})=0$ on $(0,\infty )\times {\mathbb{R}}^N$ with regular enough initial datum extinguish after some finite time $T_{\ast }>0$. More precisely, one has $\frac{u(t,\cdot )}{U_{T_{\ast },z,\hat{I}»}(t,\cdot )}-1=o(1)$ as $t\to T_{\ast }^{-}$ for a certain extinction profile $U_{T_{\ast },z,\hat{I}»}$, uniformly on ${\mathbb{R}}^N$. In this paper, we prove the quantitative bound $\frac{u(t,\cdot )}{U_{T_{\ast },z,\hat{I}»}(t,\cdot )}-1=\mathcal{O}((T_{\ast }-t{)}^{\frac{N+2s}{N-2s+2}})$, in a natural weighted energy norm. The main point here is that the exponent $\frac{N+2s}{N-2s+2}$ is sharp. This is the analogue of a recent result by Bonforte and Figalli (CPAM, 2021) valid for $s=1$ and bounded domains $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$. Our result is new also in the local case $s=1$. The main obstacle we overcome is the degeneracy of an associated linearized operator, which generically does not occur in the bounded domain setting. For a smooth bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$, we prove similar results for positive solutions to ${\partial }_{t}u+(-\hat{I}”{)}^s(u^m)=0$ on $(0,\infty )\times \hat{I}\text{c◯}$ with Dirichlet boundary conditions when $s\in (0,1)$ and $m\in (\frac{N-2s}{N+2s},1)$, under a non-degeneracy assumption on the stationary solution. An important step here is to prove the convergence of the relative error, which is new for this case.

33 pages, comments welcome!