Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain
type: Item
instance of: scholarly article
title: Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain
zbMATH Open document ID: 1178.35039
author: Theodore Kolokolnikov
author: Michael J. Ward
author: Juncheng Wei
published in: Journal of Nonlinear Science
publication date: 2009-07-24T00:00:00Z
full work available at URL: http://citeseerx.ist.psu.edu/viewdoc/summary
review text: Let \(\Omega\) be a two-dimensional domain. This paper deals with the simplified system \[ v_t=\varepsilon^2\Delta v-v+uv^2,\quad \varepsilon^2u_t=D\Delta u+a-\varepsilon^{-2}uv^2, \qquad x\in\Omega, \tag{1} \] associated with the homogeneous Neumann boundary conditions on the boundary \(\partial\Omega\), where \(0<\varepsilon\ll1\), \(D>0\), \(a>0\), are parameters. The aim of the authors is to study the behavior of spot-patterns for (1), as \(\varepsilon\) tends to zero. The one-spot solutions and the multi-spot-solutions are studied separately. In each case, a quasi-equilibrium solution is constructed, by means of matched asymptotic expansions , and its stability is investigated numerically. Some properties of the corresponding Green's functions, for the unit disk and for the rectangle, are given. Several experiments, starting with one-spot or multi-spot quasi-equilibrium solutions, in a square or in the unit disk, are presented. Some open problems are listed at the end of the paper.
reviewed by: Denise Huet
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92E20
zbMATH DE Number: 5586598
zbMATH Keywords: matched asymptotic expansions
zbMATH Keywords: Neumann Green's function
zbMATH Keywords: homogeneous Neumann boundary conditions
zbMATH Keywords: one-spot solutions
zbMATH Keywords: multi-spot-solutions
describes a project that uses: VLUGR2
describes a project that uses: CLAPACK
describes a project that uses: LAPACK
describes a project that uses: COLSYS
MaRDI profile type: Publication
OpenAlex ID: W2125467086
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DOI: 10.1007/S00332-008-9024-Z
description: scientific article