Sharp extinction rates for positive solutions of fast diffusion equations

TL;DR

This paper establishes the sharp rate at which positive solutions to fractional fast diffusion equations approach extinction, extending recent results to the fractional and local cases with new quantitative bounds.

Contribution

It provides the first sharp quantitative extinction rate for fractional fast diffusion equations, including the local case, overcoming linearized operator degeneracy.

Findings

Sharp extinction rate exponent proven to be optimal.

Quantitative bounds in weighted energy norm established.

Extension of results to local case and bounded domains.

Abstract

Let $s \in (0, 1]$ and $N > 2s$. It is known that positive solutions to the (fractional) fast diffusion equation $\partial_t u + (-\Delta)^s (u^\frac{N-2s}{N+2s}) = 0$ on $(0, \infty) \times \mathbb R^N$ with regular enough initial datum extinguish after some finite time $T_* > 0$. More precisely, one has $\frac{u(t,\cdot)}{U_{T_*, z, \lambda}(t,\cdot)} - 1 =o(1)$ as $t \to T_*^-$ for a certain extinction profile $U_{T_*, z, \lambda}$, uniformly on $\mathbb R^N$. In this paper, we prove the quantitative bound $ \frac{u(t,\cdot)}{U_{T_*, z, \lambda}(t,\cdot)} - 1 = \mathcal O( (T_*-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 $\Omega \subset \mathbb R^N$. 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 $\Omega \subset \mathbb R^N$, we prove similar results for positive solutions to $\partial_t u + (-\Delta)^s (u^m) = 0$ on $(0, \infty) \times \Omega$ 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.