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

R. Bruce Kellogg, Natalia Kopteva

Research output: Contribution to journalArticle

19 Citations (Scopus)

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.
LanguageEnglish
Pages184-208
Number of pages25
JournalJournal of Differential Equations
Volume248
Issue number1
DOIs
Publication statusPublished - 2010

Fingerprint

Reaction-diffusion Problems
Singularly Perturbed
Semilinear
Asymptotic Expansion
Supersolution
Subsolution
Multiple Solutions
Singular Perturbation
Well-posedness
Elliptic Equations
Dirichlet
Sector

Keywords

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

Cite this

Kellogg, R. Bruce ; Kopteva, Natalia. / A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain. In: Journal of Differential Equations. 2010 ; Vol. 248, No. 1. pp. 184-208.
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A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain. / Kellogg, R. Bruce; Kopteva, Natalia.

In: Journal of Differential Equations, Vol. 248, No. 1, 2010, p. 184-208.

Research output: Contribution to journalArticle

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KW - singularly perturbed

KW - semilinear

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