The Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold: Interaction of Gaussian curvature and Green's function
type: Item
instance of: scholarly article
title: The Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold: Interaction of Gaussian curvature and Green's function
zbMATH Open document ID: 1204.58026
author: W. H. Tse
author: Matthias Winter
author: Juncheng Wei
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2010-10-19T00:00:00Z
review text: Let \((S,g)\) be a compact Riemann surface without boundary. The authors look for non-trivial steady states to the Gierer-Meinhardt system (which is used to model morphogenesis in biology) on \((S,g)\) and study the weak coupling case. Various authors in the past have shown there are point condensation phenomena -- i.e., the peaks become narrow and shrink to a set of points. In the current paper, the authors consider a single spike solution on \((S,g)\) using the Lyapunov-Schmidt reduction. A convex combination of the Gaussian curvature and Green's function (and their derivatives) is linked to the peak locations and the \(o(1)\) eigenvalues. A non local eigenvalue problem determines the \(O(1)\) eigenvalues which have negative part in this case.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 92B05
zbMATH DE Number: 5802610
zbMATH Keywords: pattern formation
zbMATH Keywords: mathematical biology
zbMATH Keywords: singular perturbation
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: model morphogenesis
zbMATH Keywords: Gierer-Meinhardt system
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: Green's function
zbMATH Keywords: spiky solutions
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
full work available at URL: j.matpur.2010.03.003
OpenAlex ID: W2033046904
Wikidata QID: Q115343589
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The Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold: Interaction of Gaussian curvature and Green's function (DOI)
DOI: 10.1016/J.MATPUR.2010.03.003