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Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains. (English) Zbl 1049.35509
Summary: We use \(\Gamma\)–convergence to prove existence of stable multiple–layer stationary solutions (stable patterns) to a reaction–diffusion equation. Given nested simple closed curves in \({\mathbb R}^2\), we give sufficient conditions on their curvature so that the reaction–diffusion problem possesses a family of stable patterns. In particular, we extend to two-dimensional domains and to a spatially inhomogeneous source term, a previous result by Yanagida and Miyata.

MSC:
35K20 Initial-boundary value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
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