Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system

In this paper, we rigorously prove the existence and stability of K-peaked asymmetric patterns for the Gierer-Meinhardt system in a two-dimensional domain which are far from spatial homogeneity. We show that given any positive integers k(1), k(2)greater than or equal to1 with k(1)+k(2)=K, there are asymmetric patterns with k(1) large peaks and k(2) small peaks. Most of these asymmetric patterns are shown to be unstable. However, in a narrow range of parameters, asymmetric patterns may be stable (in contrast to the one-dimensional case). (C) 2003 Elsevier SAS. All rights reserved.