Higher-dimensional stationary solutions of a FitzHugh-Nagumo system for pattern formation in a spatially heterogeneous medium

The aim of this paper is to study pattern formation of a reaction-diffusion-ODE system with FitzHugh-Nagumo type nonlinearity in higher-dimensional domains. We construct continuous steady states, which are close to the equilibria of the kinetic system (i.e., without diffusion) by applying the sub- and super-solution method. In addition, we construct steady states with jump discontinuity via the generalized mountain pass lemma and show that they are asymptotically stable. Moreover, the existence of single transition layer solutions is proved by using the approaches of the singular perturbation method and the generalized implicit function theorems. Finally, we present some numerical simulations to illustrate the theoretical results.