The Existence and Stability of Periodic Solutions with a Boundary Layer in a Two-Dimensional Reaction-Diffusion Problem in the Case of Singularly Perturbed Boundary Conditions of the Second Kind

2020 ◽  
Vol 75 (2) ◽  
pp. 116-122 ◽  
Author(s):  
N. N. Nefedov ◽  
E. I. Nikulin
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Two Dimensional ◽  
Existence And Stability

Periodic Solutions with a Boundary Layer of Reaction–Diffusion Equations with Singularly Perturbed Neumann Boundary Conditions

2014 ◽  
Vol 24 (08) ◽  
pp. 1440019 ◽  
Author(s):  
Valentin F. Butuzov ◽  
Nikolay N. Nefedov ◽  
Lutz Recke ◽  
Klaus R. Schneider
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
Time Periodic

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 description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions.

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