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 Model. Anal. Inform. Sist., 2016, Volume 23, Number 3, Pages 342–348 (Mi mais504)

Existence and stability of periodic solutions for reaction-diffusion equations in the two-dimensional case

N. N. Nefedov, E. I. Nikulin

Lomonosov Moscow State University, GSP-1, 1-2 Leninskiye Gory, Moscow, 119991, Russia

Abstract: Parabolic singularly perturbed problems have been actively studied in recent years in connection with a large number of practical applications: chemical kinetics, synergetics, astrophysics, biology, and so on. In this work a singularly perturbed periodic problem for a parabolic reaction-diffusion equation is studied in the two-dimensional case. The case when there is an internal transition layer under unbalanced nonlinearity is considered. The internal layer is localised near the so called transitional curve. An asymptotic expansion of the solution is constructed and an asymptotics for the transitional curve is determined. The asymptotical expansion consists of a regular part, an interior layer part and a boundary part. In this work we focus on the interior layer part. In order to describe it in the neighborhood of the transition curve the local coordinate system is introduced and the stretched variables are used. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The upper and lower solutions are constructed by sufficiently complicated modification of the asymptotic expansion of the solution. The Lyapunov asymptotical stability of the solution was proved by using the method of contracting barriers. This method is based on the asymptotic comparison principle and uses the upper and lower solutions which are exponentially tending to the solution to the problem. As a result, the solution is locally unique.
The article is published in the authors' wording.

Keywords: reaction-diffusion, singular perturbations, small parameter, interior layers, unbalanced reaction, boundary layers, differential inequalities, upper and lower solutions.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-04619_à14-01-91333_ÍÍÈÎ_à This work was supported by RFBR and RFBR-DFG projects (pr. 15-01-04619, 14-01-91333).

DOI: https://doi.org/10.18255/1818-1015-2016-3-342-348

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UDC: 519
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Citation: N. N. Nefedov, E. I. Nikulin, “Existence and stability of periodic solutions for reaction-diffusion equations in the two-dimensional case”, Model. Anal. Inform. Sist., 23:3 (2016), 342–348

Citation in format AMSBIB
\Bibitem{NefNik16} \by N.~N.~Nefedov, E.~I.~Nikulin \paper Existence and stability of periodic solutions for reaction-diffusion equations in the two-dimensional case \jour Model. Anal. Inform. Sist. \yr 2016 \vol 23 \issue 3 \pages 342--348 \mathnet{http://mi.mathnet.ru/mais504} \crossref{https://doi.org/10.18255/1818-1015-2016-3-342-348} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3520856} \elib{http://elibrary.ru/item.asp?id=26246300}