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arXiv · 2105.05023

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

Szymon Cygan·Anna Marciniak-Czochra·Grzegorz Karch·Kanako Suzuki
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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)].

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Szymon Cygan, Anna Marciniak-Czochra, Grzegorz Karch, Kanako Suzuki. 2021-05-11. Instability of all regular stationary solutions to reaction-diffusion-ODE systems. https://arxiv.org/abs/2105.05023

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