The Neumann problem in an irregular domain

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Authors
Bolikowski, Łukasz
Gokieli, Maria
Varchon, Nicolas
Date
2010
Journal Title
Journal ISSN
Volume Title
Publisher
EMS, Interfaces and Free Boundaries
Abstract
We consider the stability of patterns for the reaction-diffusion equation with Neumann boundary conditions in an irregular domain in ℝN, N ≥ 2, the model example being two convex regions connected by a small ‘hole’ in their boundaries. By patterns we mean solutions having an interface, i.e. a transition layer between two constants. It is well known that in 1D domains and in many 2D domains, patterns are unstable for this equation. We show that, unlike the 1D case, but as in 2D dumbbell domains, stable patterns exist. In a more general way, we prove invariance of stability properties for steady states when a sequence of domains Ωn converges to our limit domain in the sense of Mosco. We illustrate the theoretical results by numerical simulations of evolving and persisting interfaces.
Description
Keywords
reaction-diffusion    steady states    stability    domain variation    Mosco convergence
Citation
Ł. Bolikowski, M. Gokieli and N. Varchon, The Neumann problem in an irregular domain, Interfaces and Free Boundaries, Volume 12, Issue 4, 2010, pp. 443–462.
URI
http://depot.ceon.pl/handle/123456789/68
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Artykuły ICM / ICM Articles