The Gierer-Meinhardt System with Saturation

Abstract

We consider the following shadow system of the Gierer-Meinhardt system with saturation: ⎧ ⎪ ⎨ ⎪ ⎩ A t = 2 ∆A − A + A 2 ξ(1+kA 2) in Ω × (0, ∞), τ ξ t = −ξ + 1 |Ω| Ω A 2 dx in (0, +∞), ∂A ∂ν = 0 on ∂Ω × (0, ∞), where > 0 is a small parameter, τ ≥ 0, k > 0 and Ω ⊂ R n is smooth bounded domain. The case k = 0 has been studied by many authors in recent years. Here we give some sufficient conditions on k for the existence and stability of stable spiky solutions. In the one-dimensional case we have a complete answer of the stability behavior. Central to our study are a parameterized ground-state equation and the associated nonlocal eigenvalue problem (NLEP) which is solved by functional analysis and the continuation method.

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“…For correct choice of parameters, this system of equations exhibits a Turing instability and subsequent pattern formation. A full bifurcation analysis is outside the scope of this paper, and similar studies have been performed for several variants of the GM system (Song et al, 2017; Wei and Winter, 2014). However, we do highlight sufficient conditions for the system to be unstable to spatial perturbation: in the limit γ → 0 there must exist some integer n > 0 such that one of two cases holds:…”