Clustered Spots In The FitzHugh-Nagumo System

Abstract

We construct {\bf clustered} spots for the following FitzHugh-Nagumo system: \[ \left\{ \begin{array}{l} \ep^2\Delta u +f(u)-\delta v =0\quad \mbox{in} \ \Om, \\[2mm] \Delta v+ u=0 \quad \mbox{in} \ \Om, \\[2mm] u= v =0 \quad\mbox{on} \ \partial \Om, \end{array} \right. \] where $\Om$ is a smooth and bounded domain in $R^2$. More precisely, we show that for any given integer $K$, there exists an $ \ep_{K}>0$ such that for $0<\ep <\ep_K,\, \ep^{m^{'}} \leq \delta \leq \ep^m$ for some positive numbers $m^{'}, m$, there exists a solution $(u_{\ep},v_{\ep})$ to the FitzHugh-Nagumo system with the property that $u_{\ep}$ has $K$ spikes $Q_{1}^\ep, ..., Q_K^\ep$ and the following holds:

(i) The center of the cluster $\frac{1}{K} \sum_{i=1}^K Q_i^\ep $ approaches a hotspot point $Q_0\in\Om$.

(ii) Set $l^\ep=\min_{i \not = j} |Q_i^\ep -Q_j^\ep| =\frac{1}{\sqrt{a}} \log\left(\frac{1}{\delta \ep^2 }\right) \ep ( 1+o(1))$. Then $ (\frac{1}{l^\ep} Q_1^\ep, ..., \frac{1}{l^\ep} Q_K^\ep)$ approaches an optimal configuration of the following problem:

{\it $ (*) \ \ \ $ Given $K$ points $Q_1, ..., Q_K \in R^2$ with minimum distance $1$, find out the optimal configuration that minimizes the functional $\sum_{i \not = j} \log |Q_i-Q_j|$}.