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,\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