On Unstable Solutions with a Nonmonotone Boundary Layer in a Two-Dimensional Reaction-Diffusion Problem

The paper deals with the study of time-periodic solutions of boundary layer type for a two-dimensional reaction-diffusion problem with a small parameter at the parabolic operator in the case of singularly perturbed boundary conditions of the second kind. The asymptotic approximation with respect to the small parameter for solutions with a nonmonotone boundary layer is constructed. It is shown that all such solutions are unstable. The proof of the instability of the solutions is based on the construction of an unordered pair of upper and lower solutions and on the application of a corollary of the Krein–Rutman theorem.