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

A general system of several ordinary differential equations coupled with a reaction-diffusion equation in bounded domain zero-flux boundary condition is studied the context pattern formation. These initial-boundary value problems may have regular (i.e. sufficiently smooth) stationary solutions. This class close-to-equilibrium patterns includes solutions that emerge due to Turing instability spatially constant solution. The main result this work all patterns. It suggests stable arising models non-diffusive components must be far-from-equilibrium exhibiting singularities. Such discontinuous been considered our parallel (Cygan et al., 2021 [4]).