Nonlinear stability results for stationary solutions of reaction-diffusion-ODE systems

Reaction-diffusion-ODE systems are emerging in modeling of biological pattern formation based on the coupling of diffusive and non-diffusive spatially heterogeneous processes. They may exhibit patterns with singularities such as jump-discontinuities. This work provides nonlinear stability and instability conditions for bounded stationary solutions of reaction-diffusion-ODE systems consisting of m ODEs coupled with k reaction-diffusion equations. We characterize the spectrum of the linearized operator and relate its spectral properties to the corresponding semigroup properties. Considering the function spaces $L^{\infty }{\left(\Omega \right)}^{m+k},L^{\infty }{\left(\Omega \right)}^m\times C{\left(\bar{\Omega }\right)}^k$ and $C{\left(\bar{\Omega }\right)}^{m+k}$, we establish a sign condition on the spectral bound of the linearized operator, which implies nonlinear stability or instability of the stationary pattern.