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

The semilinear reaction-diffusion equation−ε4u+b(x, u) = 0 with Dirichlet boundary conditions is considered in a convex unbounded sector. The diffusion parameter ε is arbitrarily small, and the “reduced equation” b(x, u0(x)) = 0 may have multiple solutions. A formal asymptotic expansion for a possible solution u is constructed that involves boundary and corner layer functions. For this asymptotic expansion, we establish certain inequalities that are used in [1] to construct sharp suband super-solutions and then establish the existence of a solution to a similar nonlinear elliptic problem in a convex polygon.