Stable foliations near a traveling front for reaction diffusion systems

TL;DR

This paper proves the existence of a stable foliation near traveling front solutions in reaction diffusion systems, extending stability analysis by constructing stable manifolds that foliate a neighborhood around the front.

Contribution

It introduces a method to establish stable foliations near traveling fronts in reaction diffusion equations, complementing previous orbital stability results.

Findings

Existence of stable foliation near traveling fronts

Construction of stable manifolds for small translations

Foliation of a neighborhood around the front by stable manifolds

Abstract

We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the differential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front.