Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions

We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of in...

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Bibliographic Details
Published inJournal of Differential Equations Vol. 252; no. 4; pp. 2951 - 2982
Main Authors Kučera, Milan, Väth, Martin
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.02.2012
Subjects
Degree
Global bifurcation
Neumann boundary condition
Reaction–diffusion system
Signorini condition
Stationary solutions
Variational inequality
47J20
35J60
Degree
Variational inequality
35B32
Signorini condition
Neumann boundary condition
Reaction–diffusion system
35K57
Stationary solutions
Global bifurcation
35J88
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Summary:We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2011.10.016