Boundary layer solutions in the Gierer–Meinhardt system with inhomogeneous boundary conditions

Abstract

The use of inhomogeneous boundary conditions has become increasingly important in the study of pattern forming reaction diffusion systems. Such inhomogeneities can be used to describe, for example, the interaction between an individual cell with its environment. In this paper we consider arbitrary mixed inhomogeneous boundary conditions for the activator in the singularly perturbed Gierer-Meinhardt system. By using formal asymptotics we derive an algebraic system and nonlocal eigenvalue problem respectively describing the structure and linear stability of multi-spike solutions in the case of a one-dimensional domain, thereby extending previous results obtained in the case of inhomogeneous Neumann boundary conditions by Gomez and Wei (2021). We also rigorously prove partial stability results and provide detailed stability thresholds for two examples consisting of one- and two-spike solutions. In higher dimensions we restrict our attention to the shadow limit in which the inhibitor is well-mixed and for which we rigorously establish both the existence and stability of a boundary layer solution. In both the one- and higher-dimensional cases we find that when the boundary conditions are symmetric then a symmetric solution is stable only when the magnitude of the inhomogeneity exceeds some threshold. Below this threshold we demonstrate that a stable asymmetric two spike solution emerges in the case of a one-dimensional domain, while in the case of a two-dimensional domain we show numerical simulations illustrating the formation of a near-boundary spike.