Best matching Barenblatt profiles are delayed

TL;DR

This paper investigates the asymptotic behavior of solutions to fast diffusion equations, revealing a delay effect influenced by initial data's proximity to self-similar Barenblatt solutions, measured via relative entropy.

Contribution

It introduces a novel analysis of the delay in the growth of second moments, linking it to a relative entropy measure and a best matching Barenblatt profile, enhancing understanding of solution dynamics.

Findings

Delay depends on initial data's distance to Barenblatt solutions

A new scale based on the best matching profile is monotone

The delay affects the asymptotic growth rate of moments

Abstract

The growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self-similar solutions. However, at next order, there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self-similar Barenblatt solutions. This distance can be measured in terms of a relative entropy to the best matching Barenblatt profile. This best matching Barenblatt function determines a scale. In new variables based on this scale, which are given by a self-similar change of variables if and only if the initial datum is one of the Barenblatt profiles, the typical scale is monotone and has a li