Existence and asymptotic stability of a periodic solution with boundary layers of reaction-diffusion equations with singularly perturbed Neumann boundary conditions
- Butuzov, Valentin F.
- Nefedov, Nikolai N.
- Recke, Lutz
- Schneider, Klaus
2010 Mathematics Subject Classification
- 35B25 35B35 35B12 35K57
- singularly perturbed first order ordinary differential equation, initial value problem, boundary layer, double root of degenerate equation, asymptotic expansion
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 solutions