Abstract:
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.
Key words: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.
Citation:
V. F. Butuzov, N. N. Nefedov, L. Recke, K. R. Schneider, “Existence, asymptotics, stability and region of attraction of a periodic boundary layer solution in case of a double root of the degenerate equation”, Comput. Math. Math. Phys., 58:12 (2018), 1989–2001
\Bibitem{ButNefRec18}
\by V.~F.~Butuzov, N.~N.~Nefedov, L.~Recke, K.~R.~Schneider
\paper Existence, asymptotics, stability and region of attraction of a periodic boundary layer solution in case of a double root of the degenerate equation
\jour Comput. Math. Math. Phys.
\yr 2018
\vol 58
\issue 12
\pages 1989--2001
\mathnet{http://mi.mathnet.ru/zvmmf10872}
\crossref{https://doi.org/10.1134/S0965542518120072}
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Linking options:
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This publication is cited in the following 3 articles:
I. V. Boykov, “Stability of Solutions to Systems of Parabolic Equations with Delay”, Tech. Phys., 68:3 (2023), 59
G. A. Kurina, M. A. Kalashnikova, “Singularly perturbed problems with multi-tempo fast variables”, Autom. Remote Control, 83:11 (2022), 1679–1723
A. L. Nechaeva, M. A. Davydova, “Periodic solutions with boundary layers in the problem of modeling the vertical transfer of an anthropogenic impurity in the troposphere”, Mosc. Univ. Phys. Bull., 74:6 (2019), 559–565