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

R. Bruce Kellogg, Natalia Kopteva

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23 Citations (Scopus)
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Abstract

The semilinear reaction-di®usion equation ¡"24u+b(x; u) = 0 with Dirichlet bound-ary 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 −Dz + 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
Early online date11 Sep 2009
DOIs
Publication statusPublished - 1 Jan 2010

Keywords

  • singularly perturbed
  • semilinear
  • reaction-diffusion problem
  • polygonal domain

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