Point-wise Stability of Reaction Diffusion Fronts

TL;DR

This paper establishes precise decay rates for reaction diffusion fronts using pointwise semigroup techniques, revealing both temporal exponential convergence and spatial Gaussian diffusion, with novel analytical methods.

Contribution

It introduces a new pointwise Green function decomposition and an instantaneous tracking scheme to analyze stability of reaction diffusion fronts.

Findings

Sharp decay rates in space and time for perturbed fronts

Reproduction of exponential convergence results in weighted norms

Identification of Gaussian spatial diffusion rate

Abstract

Using pointwise semigroup techniques, we establish sharp rates of decay in space and time of a perturbed reaction diffusion front to its time-asymptotic limit. This recovers results of Sattinger, Henry and others of time-exponential convergence in weighted $L^p$ and Sobolev norms, while capturing the new feature of spatial diffusion at Gaussian rate. Novel features of the argument are a point-wise Green function decomposition reconciling spectral decomposition and short-time Nash-Aronson estimates and an instantaneous tracking scheme similar to that used in the study of stability of viscous shock waves.