Instability of all regular stationary solutions to reaction-diffusion-ODE 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 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 far-from-equilibrium exhibiting singularities. Such discontinuous stationary solutions have been considered in our parallel work (Cygan et al., 2021 [4]).