A mathematical framework for determining the stability of steady states of reaction-diffusion equations with periodic source terms

TL;DR

This paper introduces a mathematical framework using Bloch's theorem and Poincaré's inequality to analyze the stability of steady states in one-dimensional reaction-diffusion equations with periodic source terms, based solely on steady state properties.

Contribution

The authors develop an a priori stability criterion for nonlinear reaction-diffusion steady states with periodic sources, applicable in one spatial dimension.

Findings

Provides a method to identify stable steady states based on steady state properties.

Framework applicable to parameter space analysis for stability regions.

Utilizes Bloch's theorem and Poincaré's inequality for stability assessment.

Abstract

We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction-diffusion equations with periodic source terms, in one spatial dimension. We formulate an \textit{a priori} condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch's theorem and Poincar\'e's inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.