Existence and asymptotic stability of periodic solutions with an interior layer of reaction–advection–diffusion equations

We consider a singularly perturbed parabolic periodic boundary value problem for a reaction–advection–diffusion equation. We construct the interior layer type formal asymptotics and propose a modified procedure to get 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. Moreover, we are able to establish the asymptotic stability of this solution.