Existence and asymptotic stability of a periodic solution with boundary layers of reaction-diffusion equations with singularly perturbed Neumann boundary conditions

Abstract

We consider singularly perturbed reaction-diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x,t,\ve) with boundary layers and derive conditions for their asymptotic stability The boundary layer part of u(x,t,\ve) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order \ve. Another peculiarity of our problem is that - in contrast to the case of Dirichlet boundary conditions - it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the desribtion of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solution