# Stable Boundary Spike Clusters for the Two-Dimensional Gierer-Meinhardt System

## TL;DR

This paper constructs and analyzes stable boundary spike clusters in a two-dimensional Gierer-Meinhardt system with small diffusivities, demonstrating their stability and the underlying mechanisms of spike interactions.

## Contribution

It introduces a method to construct stable boundary spike clusters in the Gierer-Meinhardt system with small diffusivities, highlighting the role of boundary curvature and spike interactions.

## Key findings

- Existence of boundary spike clusters approaching boundary curvature maxima
- Proof of linear stability of these spike clusters
- Identification of repulsive spike interactions due to small inhibitor diffusivity

## 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.