Existence, Asymptotics, Stability and Region of Attraction of a Periodic Boundary Layer Solution in Case of a Double Root of the Degenerate Equation
- 2 Downloads
For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.
Keywords:singularly perturbed reaction-diffusion equation double root of the degenerate equation initial boundary value problem asymptotic expansion asymptotically stable periodic solution region of attraction
This work is supported by RFBR, pr. N 16-01-00437, N 15-01-04619, by the DFG grant RE1336/1-1 and by the program of cooperation of the Moscow State University and the Humboldt University of Berlin.
- 1.P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Math, Ser. 247 (Longman Scientific & Technical, Harlow, 1991).Google Scholar
- 7.V. F. Butuzov, N. N. Nefedov, L. Recke, and K. R. Schneider, “Periodic solutions with a boundary layer of reaction-diffusion equations with singularly perturbed Neumann boundary conditions,” Int. J. Bif. Chaos 24 (8), Article ID 1440019 (2014).Google Scholar