Reaction-diffusion equations coupled to ordinary differential equations (ODEs) may exhibit spatially lowregular 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 ∞ (Ω) m+k , L ∞ (Ω) m × C(Ω) k and C(Ω) 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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