2013, Butuzov, Valentin F., Nefedov, Nikolai N., Recke, Lutz, Schneider, Klaus R.

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; epsilon) with boundary layers and derive conditions for their asymptotic stability The boundary layer part of u(x; t; ") is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order epsilon.

2012, Nefedov, Nikolai N., Recke, Lutz, Schneider, Klaus R.

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 with interior layer.