Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods

TL;DR

This paper analyzes the convergence rates of solutions to the fast diffusion equation on bounded domains, using energy methods to establish stability and asymptotic profiles, including sign-changing solutions.

Contribution

It provides an energy-based approach to quantify convergence rates to asymptotic profiles, offering an alternative proof to existing entropy-based results and exploring sign-changing solution dynamics.

Findings

Established sharp convergence rates to positive profiles.

Proved exponential stability of sign-changing profiles in dumbbell domains.

Extended analysis to solutions with changing signs under specific conditions.

Abstract

This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles will be revealed via an energy method. The sharp rate of convergence to \emph{positive} ones was recently discussed by Bonforte and Figalli (2021, CPAM) based on an entropy method. An alternative proof for their result will also be provided. Furthermore, dynamics of fast diffusion flows with changing signs will be discussed more specifically under concrete settings; in particular, exponential stability of some sign-changing asymptotic profiles will be proved in dumbbell domains for initial data with certain symmetry.