On the Existence and Asymptotic Stability of Two-Dimensional Periodic Solutions with an Internal Transition Layer in a Problem with a Finite Advection

We consider a periodic boundary value problem for a singularly perturbed reaction-advection-diffusion equation in the case of a two-dimensional space variable. We construct a new interior layer-type formal asymptotics which includes an approximation of the location of the interior layer, investigate the order-preserving properties of the operators generating the asymptotics, and propose a modified procedure to obtain asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. We also prove the asymptotic Lyapunov stability of this solution.

DOI 10.1134/S1061920824040113