Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and the quasi-invariant manifold
type: Item
instance of: scholarly article
title: Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and the quasi-invariant manifold
zbMATH Open document ID: 0962.35063
DOI: 10.1215/S0012-7094-99-09802-2
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The author considers the problem: \(-\Delta u+ \varepsilon^{-2}(u- f(u))= 0\), in \(\Omega\), \(\partial u/\partial n= 0\) on \(\partial\Omega\), \(u>0\) in \(\Omega\), where \(\varepsilon\) is a small parameter. \(\Omega\subset \mathbb{R}^d\) is a bounded domain with \(C^3\) boundary and \(f\in C^{1,1}(\mathbb{R})\) satisfies \(f\equiv 0\) in \((-\infty, 0)\), \(f(u)= O(u^p)\) as \(u\to\infty\), and \(1< p<(d+ 2)/(d- 2)\), (if \(d=2\), then \(1< p<\infty\)). Additionally, it assumes that there exists \(q>1\) such that for \(u>0\) the function \(f(u)/u^q\) is nondecreasing.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 92C15
zbMATH DE Number: 1425216
zbMATH Keywords: multiple spike layer
zbMATH Keywords: Gierer-Meinhardt system
zbMATH Keywords: quasi-invariant manifold
author: Michal Kowalczyk
reviewed by: Jiaqi Mo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1425216