A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain

R. Bruce Kellogg, Natalia Kopteva

Research output: Contribution to journalArticlepeer-review

Abstract

The semilinear reaction-diffusion equation - ε2 Δ u + b (x, u) = 0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the "reduced equation" b (x, u0 (x)) = 0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation - Δ z + f (z) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.

Original languageEnglish
Pages (from-to)184-208
Number of pages25
JournalJournal of Differential Equations
Volume248
Issue number1
DOIs
Publication statusPublished - 1 Jan 2010

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