Boundary spikes in the Gierer-Meinhardt system
type: Item
instance of: scholarly article
title: Boundary spikes in the Gierer-Meinhardt system
zbMATH Open document ID: 1163.35354
author: P. Felmer
author: Manuel del Pino
author: Michal Kowalczyk
published in: Communications on Pure and Applied Analysis
publication date: 2003-04-08T00:00:00Z
review text: The authors consider the following system of reaction-diffusion equations \[ \begin{cases} a_t-\varepsilon^2\Delta a+a-a^2/h=0 &\text{in }\Omega\times (0,\infty),\\ h_t-D\Delta h+h-a^2=0 &\text{in }\Omega\times (0,\infty),\\ \partial_na=\partial_nh=0 &\text{on }\partial\Omega\times (0,\infty),\\ a(\cdot,0)= a_0>0, \;h(\cdot,0)=h_0>0, &{}\end{cases} \tag{1} \] where \((a,h)\) represents the concentration of the activator and inhibitor respectively, and the parameters \(\varepsilon^2\), \(D>0\) are their diffusion rates, \(\Omega\subset\mathbb R^N\), \(N\leq 3\) is a bounded domain with smooth boundary, \(\partial_n\) denotes the exterior normal derivative. The main goal of this paper is to show the existence of steady state solutions to (1) for the range of \(\varepsilon\) and \(D\).
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1894767
zbMATH Keywords: steady state
zbMATH Keywords: boundary spikes
zbMATH Keywords: Gierer-Meinhardt system
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: cpaa.2002.1.437
OpenAlex ID: W1992561852
DOI: 10.3934/CPAA.2002.1.437
description: scientific article; zbMATH DE number 1894767