Existence and Stability of a Boundary Layer with an Interior Spike in the Singularly Perturbed Shadow Gierer-Meinhardt System

TL;DR

This paper analyzes the existence and stability of boundary layer with interior spike solutions in a singularly perturbed Gierer-Meinhardt system, revealing conditions for their stability and constructing solutions using asymptotic methods.

Contribution

It introduces a detailed asymptotic analysis of boundary layer spike solutions in the GM system, identifying stable and unstable types and their parameter ranges.

Findings

Two types of boundary layer spike solutions are constructed.

One solution type is unconditionally linearly stable.

Both solutions are stable over a large range of activator flux values.

Abstract

The singularly perturbed Gierer-Meinhardt (GM) system in a bounded $d$-dimensional domain ($d\geq 2$) is known to exhibit boundary layer (BL) solutions for a non-zero activator flux. It was previously shown that such BL solutions can be destabilized by decreasing the activator flux below a stability threshold. Moreover, numerical simulations previously indicated that solutions consisting of a boundary layer and interior spike emerge after the destabilization of a BL solution. In this paper we use the method of matched asymptotic expansions to investigate the structure and stability of such "boundary layer spike" (BLS) solutions in the presence of an asymptotically small activator diffusivity $\varepsilon^2\ll 1$. We find that two types of BLS solutions, one of which is unconditionally linearly stable and the other unstable, can be constructed provided that the activator flux is sufficiently small. In this way we determine that there is an asymptotically large range of activator flux values for which both the BL solution and one of the BLS solutions are linearly stable. Formal asymptotic calculations are further validated by numerically simulating the singularly perturbed GM system.