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 −∆ is introduced to study the behavior of the regular part of Green's function around the boundary. This reflection generalizes the wellknown notions of mirror image and circle inversion for domains with sufficiently smooth boundary.