Extinction profiles for the Sobolev critical fast diffusion equation in bounded domains. I. One bubble dynamics

TL;DR

This paper studies how solutions to a Sobolev critical fast diffusion equation in bounded domains either stabilize or blow up, depending on initial conditions, revealing detailed extinction dynamics and bubble behavior.

Contribution

It introduces a detailed analysis of extinction profiles and bubble dynamics for the Sobolev critical fast diffusion equation in bounded domains.

Findings

Solutions converge to steady states or blow-up bubbles under certain initial conditions.

The paper characterizes the dichotomy in the extinction behavior of solutions.

Provides a framework for understanding bubble formation and extinction in fast diffusion equations.

Abstract

In this paper, we investigate the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble.