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|Title:||Stable boundary spike clusters for the two-dimensional Gierer-Meinhardt system|
|Keywords:||Pattern formation;Reaction-diffusion system;Spike;Cluster;Stability|
|Citation:||Journal de Mathématiques Pures et Appliquées|
|Abstract:||We consider the Gierer-Meinhardt system with small inhibitor diﬀusivity and very small activator diﬀusivity in a bounded and smooth two-dimensional domain. For any given positive integer k we construct a spike cluster consisting of k boundary spikes which all approach the same nondegenerate local maximum point of the boundary curvature. We show that this spike cluster is linearly stable. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diﬀusivity the interaction between spikes is repulsive and the spikes are attracted towards a nondegenerate local maximum point of the boundary curvature. Combining these two eﬀects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.|
|Appears in Collections:||Dept of Mathematics Embargoed Research Papers|
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