The FitzHugh-Nagumo equations have been studied from various viewpoints such as traveling pulses, pattern formation, and so on. In this talk we study the case where the activator does not diffuse. This may be regarded as a reference system for pattern formation in ODE-RD systems. Our fundamental interest lies in understanding the relationship between the initial disturbance and the final pattern.
For this purpose, we reduce the system as simple as possible by taking the limit of large diffusion rate of the inhibitor and furthermore by considering the quasi-stationary approximation for the inhibitor. We report some rigorous results on the long time behavior of solutions of the initial-boundary value problem for this simplified model, restricting ourselves to special class of initial functions (joint work with Ying Li).