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

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.