Steady states of fitzhugh-nagumo system with non-diffusive activator and diffusive inhibitor

In this paper, we consider a diffusion equation coupled to an ordinary differential equation with FitzHugh-Nagumo type nonlinearity. We construct continuous spatially heterogeneous steady states near, as well as far from, constant steady states and show that they are all unstable. In addition, we construct various types of steady states with jump discontinuities and prove that they are stable in a weak sense defined by Weinberger. The results are quite different from those for classical reaction-diffusion systems where all species diffuse.