On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system

TL;DR

This paper investigates the dynamics of a non-local parabolic equation derived from the Gierer-Meinhardt system, focusing on blow-up phenomena and pattern formation due to diffusion-driven instability.

Contribution

It provides new insights into the blow-up behavior and pattern formation mechanisms in a non-local equation related to the Gierer-Meinhardt model, especially under weak inhibitor response.

Findings

Derivation of blow-up results indicating instability patterns.

Identification of diffusion-driven (Turing) instability near stationary solutions.

Evidence of stable patterns from global-in-time solution analysis.

Abstract

The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.