Stability Results for Bounded Stationary Solutions of Reaction-Diffusion-ODE Systems

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.