Behaviour near extinction for the Fast Diffusion Equation on bounded domains

TL;DR

This paper investigates the behavior of solutions to the Fast Diffusion Equation near the extinction time on bounded domains, establishing uniform convergence and rates of convergence using entropy methods and elliptic problem analysis.

Contribution

It provides a detailed description of the solution behavior near extinction, including uniform convergence in relative error and convergence rates for parameters close to 1.

Findings

Solutions converge uniformly in relative error norm near extinction

Explicit rates of convergence are obtained for $m$ close to 1

Analysis of stationary elliptic problem properties is essential for the results

Abstract

We consider the Fast Diffusion Equation $u_t=\Delta u^m$ posed in a bounded smooth domain $\Omega\subset \RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positive solutions $u(t,x)$ of such problem extinguish in a finite time $T$, and also that such solutions approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. Here we are interested in describing the behaviour of the solutions near the extinction time. We first show that the convergence $u(t,x)\,(T-t)^{-1/(1-m)}$ to $S(x)$ takes place uniformly in the relative error norm. Then, we study the question of rates of convergence of the rescaled flow. For $m$ close to 1 we get such rates by means of entropy methods and weighted Poincar\'e inequalities. The analysis of the latter point makes an essential use of fine properties of the associated stationary elliptic problem $-\Delta S^m= {\bf c} S$ in the limit $m\to 1$, and such a study has an independent interest.