Fast diffusion equations: matching large time asymptotics by relative entropy methods

TL;DR

This paper introduces a novel approach using relative entropy and a dynamic coordinate change based on second moments to improve the large-time asymptotic analysis of solutions to the fast diffusion equation, surpassing previous methods.

Contribution

It presents a new non self-similar change of coordinates and entropy method that better matches the asymptotics of solutions for large times in the fast diffusion equation.

Findings

Enhanced asymptotic matching for large-time solutions.

Method applicable when Barenblatt solutions have finite second moments.

Improved convergence estimates compared to traditional self-similar approaches.

Abstract

A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.