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Stability Results for Bounded Stationary Solutions of Reaction-Diffusion-ODE Systems (2201.12748v2)

Published 30 Jan 2022 in math.AP and math.SP

Abstract: Reaction-diffusion equations coupled to ordinary differential equations (ODEs) may exhibit spatially low-regular stationary solutions. This work provides a comprehensive theory of asymptotic stability of bounded, discontinuous or continuous, stationary solutions of reaction-diffusion-ODE systems. We characterize the spectrum of the linearized operator and relate its spectral properties to the corresponding semigroup properties. Considering the function spaces $L\infty(\Omega){m+k}, L\infty(\Omega)m \times C(\overline{\Omega})k$ and $C(\overline{\Omega}){m+k}$, we establish a sign condition on the spectral bound of the linearized operator, which implies nonlinear stability or instability of the stationary pattern.

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