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Mat. Zametki, 2019, Volume 106, Issue 5, Pages 708–722 (Mi mz12335)  

Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection

N. N. Nefedov, E. I. Nikulin

Lomonosov Moscow State University

Abstract: We consider the boundary-value singularly perturbed time-periodic problem for the parabolic reaction-advection-diffusion equation in the case of a weak linear advection in a two-dimensional domain. The main result of the present paper is the justification, under certain sufficient assumptions, of the existence of a periodic solution with internal transition layer near some closed curve and the study of the Lyapunov asymptotic stability of such a solution. For this purpose, an asymptotic expansion of the solution is constructed; the justification of the existence of the solution with the constructed asymptotics is carried out by using the method of differential inequalities. The proof of Lyapunov asymptotic stability is based on the application of the so-called method of contraction barriers.

Keywords: singularly perturbed parabolic problem, reaction-advection-diffusion equations, periodic contrast structures.

Funding Agency Grant Number
Russian Science Foundation 18-11-00042
This work was supported by the Russian Science Foundation under grant 18-11-00042.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm12335

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English version:
Mathematical Notes, 2019, 106:5, 771–783

Bibliographic databases:

UDC: 517.9
Received: 12.03.2019

Citation: N. N. Nefedov, E. I. Nikulin, “Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection”, Mat. Zametki, 106:5 (2019), 708–722; Math. Notes, 106:5 (2019), 771–783

Citation in format AMSBIB
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\by N.~N.~Nefedov, E.~I.~Nikulin
\paper Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection
\jour Mat. Zametki
\yr 2019
\vol 106
\issue 5
\pages 708--722
\mathnet{http://mi.mathnet.ru/mz12335}
\crossref{https://doi.org/10.4213/mzm12335}
\transl
\jour Math. Notes
\yr 2019
\vol 106
\issue 5
\pages 771--783
\crossref{https://doi.org/10.1134/S0001434619110105}
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