2023/08/18 by Daniel Gomez, Juncheng Wei, Gomez, Daniel +1
Computer Science · Engineering · Mathematics · #35B25 #35B30 (Secondary) #35B36 (Primary) #35K57 #Differential Equations and Numerical Methods #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.2308.09620
openalex publication_date 2023/08/18 · openalex created_date 2023/08/22 · openalex updated_date 2026/07/28
The singularly perturbed Gierer-Meinhardt (GM) system in a bounded d-dimensional domain (d≥ 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 ε2≪ 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.