Instability of all regular stationary solutions to reaction-diffusion-ODE systems

TL;DR

This paper proves that all regular stationary solutions in certain reaction-diffusion-ODE systems are unstable, implying stable patterns must be far-from-equilibrium and possibly singular, which challenges the stability of near-equilibrium patterns.

Contribution

It demonstrates the universal instability of regular stationary solutions in coupled reaction-diffusion-ODE systems, highlighting the necessity of singular solutions for stability.

Findings

All regular stationary solutions are unstable.

Stable patterns require far-from-equilibrium singular solutions.

Near-equilibrium solutions cannot be stable in these systems.

Abstract

A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value problems may have regular (i.e. sufficiently smooth) stationary solutions. This class of {\it close-to-equilibrium} patterns includes stationary solutions that emerge due to the Turing instability of a spatially constant stationary solution. The main result of this work is instability of all regular patterns. It suggests that stable stationary solutions arising in models with non-diffusive components must be {\it far-from-equilibrium} exhibiting singularities. Such discontinuous stationary solutions have been considered in our parallel work [\textit{Stable discontinuous stationary solutions to reaction-diffusion-ODE systems}, preprint (2021)].