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

We consider the Gierer–Meinhardt system with precursor inhomogeneity and two small diffusivities in an interval $$\begin{equation*} \left\{ \begin{array}{ll} A_t=\epsilon^2 A''- \mu(x) A+\frac{A^2}{H}, &x\in(-1, 1),\,t>0,\\[3mm] \tau H_t=D H'' -H+ A^2, & x\in (-1, 1),\,t>0,\\[3mm] A' (-1)= A' (1)= H' (-1) = H' (1) =0, \end{array} \right. \end{equation*}$$ $$\begin{equation*}\mbox{where } \quad 0<\epsilon \ll\sqrt{D}\ll 1, \quad \end{equation*}$$ $$\begin{equation*} \tau\geq 0 \mbox{ and $\tau$ is independent of $\epsilon$. } \end{equation*}$$ A spike cluster is the combination of several spikes which all approach the same point in the singular limit. We rigorously prove the existence of a steady-state spike cluster consisting of N spikes near a non-degenerate local minimum point t 0 of the smooth positive inhomogeneity μ( x ), i.e. we assume that μ′( t 0 ) = 0, μ″( t 0 ) > 0 and we have μ( t 0 ) > 0. Here, 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: the O (1) scale of the precursor inhomogeneity μ( x ), the $O(\sqrt{D})$ scale of the inhibitor diffusivity and the O (ε) scale of the activator diffusivity; (iii) the expressions can be made explicit and often have a particularly simple form.

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