Stability Results for Bounded Stationary Solutions of Reaction-Diffusion-ODE Systems

TL;DR

This paper develops a comprehensive stability theory for bounded stationary solutions of reaction-diffusion-ODE systems, linking spectral properties of the linearized operator to the stability of spatial patterns.

Contribution

It characterizes the spectrum of the linearized operator and relates spectral bounds to nonlinear stability or instability of stationary solutions in various function spaces.

Findings

Spectral bounds determine stability or instability of stationary solutions.

A sign condition on the spectral bound implies nonlinear stability.

The theory applies to discontinuous and continuous stationary solutions.

Abstract

Reaction-diffusion equations coupled to ordinary differential equations (ODEs) may exhibit spatially low-regular stationary solutions. This work provides a comprehensive theory of asymptotic stability of bounded, discontinuous or continuous, stationary solutions of reaction-diffusion-ODE systems. We characterize the spectrum of the linearized operator and relate its spectral properties to the corresponding semigroup properties. Considering the function spaces $L^\infty(\Omega)^{m+k}, L^\infty(\Omega)^m \times C(\overline{\Omega})^k$ and $C(\overline{\Omega})^{m+k}$, we establish a sign condition on the spectral bound of the linearized operator, which implies nonlinear stability or instability of the stationary pattern.