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

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, ε) with boundary layers and derive conditions for their asymptotic stability The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order ε. 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 solutions. 1 Statement of the problem. We consider the following singularly perturbed parabolic periodic boundary value problem with singularly perturbed Neumann conditions Nε(u) := ε 2 ( ∂u ∂x2 − ∂u ∂t ) − f(u, x, t, ε) = 0 for (x, t) ∈ D := {(x, t) ∈ R : −1 < x < 1, t ∈ R}, ε ∂u ∂x (−1, t, ε) = u(−)(t), ε ∂u ∂x (1, t, ε) = u(t) for t ∈ R, u(x, t, ε) = u(x, t+ T, ε) for t ∈ R, −1 ≤ x ≤ 1 (1.1) for ε ∈ Iε0 := {0 < ε ≤ ε0}, 0 < ε0 1, f , u(−) and u are sufficiently smooth and T -periodic in t. Our interest in such problems is motivated by reaction-diffusion problems with a strong flow on the boundary. This fact is described in the paper by Nesterov [1], where a linear reactiondiffusion equation was considered. Another motivation comes from the study of singularly perturbed problems with multiple roots of the degenerate equation. We illustrate this fact by the following problem with a double root of