Clustered spots in the FitzHugh-Nagumo system

We construct clustered spots for the following FitzHugh-Nagumo system: {epsilon(2) Delta n + f (u) - delta v = 0 in Omega {Delta n + u= 0 in Omega {u = v = 0 on partial derivative Omega, where Omega is a smooth and bounded domain in R-2. More precisely, we show that for any given integer K, there exists an epsilon(K) > 0 such that for 0 < epsilon < epsilon(K), epsilon(m') <=, delta <= epsilon(m) or some positive numbers m', m, there exists a solution (u(epsilon), v(epsilon)) to the FitzHugh-Nagumo system with the property that u(epsilon) has K spikes Q(I)(epsilon)...Q(K)(epsilon) and the following holds: (i) The center of the cluster I/K Sigma(K)(i=1) Q(i)(epsilon) approaches a hotspot point Q(0) epsilon Omega. (ii) Set I-epsilon = min(i=j) vertical bar Q(i)(epsilon) - Q(j)(epsilon)vertical bar = 1/root a log (1/delta epsilon(2)) epsilon(1 + o(1)). Then (1/l(epsilon)Q(i)(epsilon),..., 1/l(K)(epsilon)) approaches an optimal configuration of the following problem: (*) Given K points Q(1),...,Q(K) epsilon R-2 with minimum distance 1, find out the optimal configuration that minimizes the functional Sigma(i not equal j) log vertical bar Q(i) - Q(j)vertical bar. (c) 2004 Elsevier Inc. All rights reserved.