Some asymptotic expansions for a semilinear reaction-diffusion problem in a sector

TL;DR

This paper develops asymptotic expansions for a semilinear reaction-diffusion problem in a sector, addressing boundary and corner layers, and sets the stage for proving solution existence in related nonlinear elliptic problems.

Contribution

It introduces formal asymptotic expansions involving boundary and corner layers for a reaction-diffusion problem in a sector, aiding in solution existence proofs.

Findings

Constructed formal asymptotic expansion with boundary and corner layer functions

Established inequalities for the asymptotic expansion

Lays groundwork for solution existence in nonlinear elliptic problems

Abstract

A semilinear singularly perturbed reaction-diffusion equation with Dirichlet boundary conditions is considered in a convex unbounded sector. The singular perturbation parameter is arbitrarily small, and the "reduced equation" may have multiple solutions. A formal asymptotic expansion for a possible solution is constructed that involves boundary and corner layer functions. For this asymptotic expansion, we establish certain inequalities that are used in a subsequent paper to construct sharp sub- and super-solutions and then establish the existence of a solution to a similar nonlinear elliptic problem in a convex polygon.