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A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial boundary value problems may have different types of stationary solutions. In our parallel work [Instability of all regular stationary solutions to reaction-diffusion-ODE systems (2021)], regular (i.e. sufficiently smooth) stationary solutions are shown to exist, however, all of them are unstable. The goal of this work is to construct discontinuous stationary solutions to general reaction-diffusion-ODE systems and to find sufficient conditions for their stability. [ABSTRACT FROM AUTHOR]

Abstract: Motivated by numerical simulations showing the emergence of either periodic or irregular patterns, we explore a mechanism of pattern formation arising in the processes described by a system of a single reaction–diffusion equation coupled with ordinary differential equations. We focus on a basic model of early carcinogenesis proposed by Marciniak-Czochra and Kimmel [Comput. Math. Methods Med. 7 (2006) 189–213], [Math. Models Methods Appl. Sci. 17 (suppl.) (2007) 1693–1719], but the theory we develop applies to a wider class of pattern formation models with an autocatalytic non-diffusing component. The model exhibits diffusion-driven instability (Turing-type instability). However, we prove that all Turing-type patterns, i.e., regular stationary solutions, are unstable in the Lyapunov sense. Furthermore, we show existence of discontinuous stationary solutions, which are also unstable. [Copyright &y& Elsevier]

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]). [ABSTRACT FROM AUTHOR]

We consider an initial-boundary value problem for reaction-diffusion equations coupled with the Keller-Segel system from the chemotaxis theory which describe a formation of colony patterns of bacteria Escherichia coli. The main goal of this work is to show that global-in-time solutions of this model converge towards stationary solutions depending on initial conditions. [ABSTRACT FROM AUTHOR]