Stable spike clusters for the one-dimensional Gierer-Meinhardt system

We consider the Gierer-Meinhardt system with precursor inhomogeneity in a one-dimensional interval. A spike cluster is the combination of several spikes which all approach the same point in the singular limit of small activator diffusivity. We rigorously prove the existence of a steady-state spike cluster consisting of N spikes near a non-degenerate local minimum point of the smooth inhomogeneity, where N is an arbitrary positive integer. Further, we show that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of order O(1)) and small (of order o(1)). The main features of studying the Gierer-Meinhardt system in this setting are as follows: (i) it is biologically relevant since it models a hierarchical process (pattern formation of small-scale structures induced by a pre-existing large-scale inhomogeneity), (ii) it contains three different spatial scales two of which are small. (iii) all the expressions can be made explicit and often have a particularly simple form.