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

Kellogg, R. Bruce and Kopteva, Natalia (2010) A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain. Journal of Differential Equations, 248 (1). pp. 184-208. ISSN 0022-0396

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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.