Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation

TL;DR

This paper investigates how solutions to the fast diffusion equation approach self-similar profiles near extinction, revealing a range of convergence rates influenced by initial data decay.

Contribution

It introduces a continuum of convergence rates for solutions approaching Barenblatt profiles, explicitly linked to initial data decay rates.

Findings

Identifies a continuum of convergence rates

Shows rates depend on initial data decay

Provides explicit formulas for convergence rates

Abstract

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data.