There are some mathematical models of a pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with an ordinary differential equation (reaction-diffusion-ODE system). Such models arise when studying coupling of the diffusive processes with processes which are localized in space, such as, for example, growth processes or intracellular signaling. This type of models exhibits the diffusion-driven instability, which has been often used to explain de novo pattern formation. The dynamics of reaction-diffusion-ODE systems appear to be very different from that of classical reaction-diffusion models. In this paper, we consider some reaction-diffusion-ODE systems, and show that a certain natural (autocatalysis) property of systems leads to instability of all inhomogeneous stationary solutions. Moreover, we discuss a possible large time behavior of solutions. We will see that space inhomogeneous solutions of the problem become unbounded in either finite or infinite time, even if space homogeneous solutions are bounded uniformly in time.
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