Stability of nonconstant steady states in reaction-diffusion systems on graphs

Diffusion process on a network of thin media can be described by a system of parabolic PDEs on a graph which interact with each other through connecting nodes. The aim of this paper is to study the existence and non-existence of stable nonconstant steady states in reaction-diffusion systems on graphs. It is shown that there are five types of graphs on which any stable steady state is necessarily constant. It is also shown that if the graph is of none of these five types, then a stable nonconstant steady state can exist by taking f appropriately.