Interior and boundary spikes for the two-dimensional Gierer–Meinhardt system

Abstract

We prove the existence of assemblies of interior and boundary spikes as stationary solutions of the two-dimensional Gierer–Meinhardt system. Moreover, we also discover that the locations of the spikes are determined by the curvature of the domain boundary together with the Green's function of the domain. A reflection operator of −Δ is introduced to study the behaviour of the regular part of Green's function around the boundary. This reflection generalizes the well-known notions of mirror image and circle inversion for domains with sufficiently smooth boundary.