Asymptotics near extinction for nonlinear fast diffusion on a bounded domain

TL;DR

This paper analyzes the asymptotic behavior near extinction for nonlinear fast diffusion on bounded domains, providing precise convergence rates and refining previous conjectures and results.

Contribution

It offers a detailed quantification of convergence rates to extinction profiles, including exponential and algebraic rates, and introduces a simplified approach accommodating zero modes.

Findings

Convergence to extinction profile is either exponential or algebraic.

Refines Berryman and Holland's 1980 conjecture on decay rates.

Provides a new method handling non-isolated vanishing profiles with zero modes.

Abstract

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).