Rate of Convergence to Separable Solutions of the Fast Diffusion Equation

TL;DR

This paper investigates how positive solutions of the fast diffusion equation approach separable solutions near extinction, establishing stability and optimal convergence rates for a specific class of solutions.

Contribution

It introduces a class of functions within the domain of attraction of separable solutions and proves their stability and optimal convergence rates.

Findings

Separable solutions are stable in a suitable sense.

Established optimal rates of convergence to separable solutions.

Identified a class of functions that belong to the domain of attraction.

Abstract

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a subcritical exponent. We show that separable solutions are stable in some suitable sense by finding a class of functions which belong to their domain of attraction. For solutions in this class we establish optimal rates of convergence to separable solutions.