Bubbling and extinction for some fast diffusion equations in bounded domains

TL;DR

This paper analyzes the extinction behavior and convergence rates of solutions to Sobolev critical and subcritical fast diffusion equations in bounded domains, revealing polynomial and exponential decay patterns.

Contribution

It provides new extinction profiles and decay rate results for fast diffusion equations with Brezis-Nirenberg effect, using novel regularity and blow-up analysis techniques.

Findings

Polynomial convergence rates for Sobolev critical case

Exponential decay for generic domains

Extinction profiles of positive solutions

Abstract

We study a Sobolev critical fast diffusion equation in bounded domains with the Brezis-Nirenberg effect. We obtain extinction profiles of its positive solutions, and show that the convergence rates of the relative error in regular norms are at least polynomial. Exponential decay rates are proved for generic domains. Our proof makes use of its regularity estimates, a curvature type evolution equation, as well as blow up analysis. Results for Sobolev subcritical fast diffusion equations are also obtained.