Cluster solutions for the FitzHugh-Nagumo system with Neumann boundary conditions

Abstract

We study cluster solution of the following FitzHugh-Nagumo system:{ε2Δu+f(u)−v=0in Ω,Δv−γv+βu=0in Ω,∂u∂n=∂v∂n=0on ∂Ω, where Ω is a smooth bounded domain in R2 and ε,γ,β are positive parameters. Under appropriate parameter regimes, given any integer k≥2, this system admits a cluster solution uε with k-boundary spikes that accumulate at Q0, a nondegenerate local maximum point of boundary curvature, whenever the term βεγ is sufficiently large or sufficiently small. Moreover, we show that these k-boundary spikes are separated from each other at a distance of O(log⁡(βεγ)γ) as βεγ→∞ or O((βε)12) as βεγ→0, respectively. The latter case proposes a new concentration feature of the activator-inhibitor type reaction-diffusion system which is barely observed in Gierer-Meinhardt system. The mechanism behind the scene is driven by two competing forces: a short range gravitational force with mass positioned at a local maximum point of the curvature and a long range repelling force between spikes whose major part comes from the Green's function of screened Poisson operator on the half space.