Stability of the steady states in multidimensional reaction diffusion systems arising in combustion theory

TL;DR

This paper proves the asymptotic stability of steady states in certain multidimensional reaction-diffusion systems, especially those relevant to combustion theory, by analyzing their behavior in weighted Sobolev spaces.

Contribution

It establishes stability results for steady states in multidimensional reaction-diffusion systems, focusing on combustion-related equations and their relation to traveling front end states.

Findings

Steady states are asymptotically stable in weighted Sobolev spaces.

Results apply to systems arising in combustion theory.

Complement previous work on traveling front stability.

Abstract

We prove that the steady state of a class of multidimensional reaction-diffusion systems is asymptotically stable at the intersection of unweighted space and exponentially weighted Sobolev spaces, and pay particular attention to a special case, namely, systems of equations that arise in combustion theory. The steady-state solutions considered here are the end states of the traveling fronts associated with the systems, and thus the present results complement recent papers \cite{GLS1, GLS2, GLS3, GLSR, GLY} that study the stability of traveling fronts.