Multi-Spike Patterns in the Gierer-Meinhardt System with a Non-Zero Activator Boundary Flux

TL;DR

This paper studies how inhomogeneous boundary conditions influence spike formation, stability, and pattern diversity in the Gierer-Meinhardt reaction-diffusion system, revealing increased stability and existence ranges for certain spike patterns.

Contribution

It introduces a new analysis of boundary layer spikes under inhomogeneous boundary conditions and develops shifted Nonlocal Eigenvalue Problems with partial stability results.

Findings

Inhomogeneous boundary conditions induce boundary layer spikes near domain edges.

These conditions expand the parameter range for stable asymmetric two-spike patterns.

The study combines asymptotic, rigorous, and numerical methods for analysis.

Abstract

The structure, linear stability, and dynamics of localized solutions to singularly perturbed reaction-diffusion equations has been the focus of numerous rigorous, asymptotic, and numerical studies in the last few decades. However, with a few exceptions, these studies have often assumed homogeneous boundary conditions. Motivated by the recent focus on the analysis of bulk-surface coupled problems we consider the effect of inhomogeneous Neumann boundary conditions for the activator in the singularly perturbed one-dimensional Gierer-Meinhardt reaction-diffusion system. We show that these boundary conditions necessitate the formation of spikes that concentrate in a boundary layer near the domain boundaries. Using the method of matched asymptotic expansions we construct boundary layer spikes and derive a new class of shifted Nonlocal Eigenvalue Problems for which we rigorously prove partial stability results. Moreover by using a combination of asymptotic, rigorous, and numerical methods we investigate the structure and linear stability of selected one- and two-spike patterns. In particular we find that inhomogeneous Neumann boundary conditions increase both the range of parameter values over which asymmetric two-spike patterns exist and are stable.