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 Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 3, Pages 21–44 (Mi izv8478)

Asymptotic behaviour and stability of solutions of a singularly perturbed elliptic problem with a triple root of the degenerate equation

V. F. Butuzov

Lomonosov Moscow State University, Faculty of Physics

Abstract: We construct and justify asymptotic expansions of solutions of a singularly perturbed elliptic problem with Dirichlet boundary conditions in the case when the corresponding degenerate equation has a triple root. In contrast to the case of a simple root, the expansion is with respect to fractional (non-integral) powers of the small parameter, the boundary-layer variables have another scaling, and the boundary layer has three zones. This gives rise to essential modifications in the algorithm for constructing the boundary functions. Solutions of the elliptic problem are stationary solutions of the corresponding parabolic problem. We prove that such a stationary solution is asymptotically stable and find its global domain of attraction.

Keywords: singularly perturbed elliptic problem, multiple root of the degenerate equation, three-zone boundary layer, stability of stationary solutions.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-04619 This paper was written with the support of RFBR (grant no. 15-01-04619).

DOI: https://doi.org/10.4213/im8478

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English version:
Izvestiya: Mathematics, 2017, 81:3, 481–504

Bibliographic databases:

UDC: 519.632.34
MSC: 35B25, 35J91, 35K20, 35K58
Revised: 02.04.2016

Citation: V. F. Butuzov, “Asymptotic behaviour and stability of solutions of a singularly perturbed elliptic problem with a triple root of the degenerate equation”, Izv. RAN. Ser. Mat., 81:3 (2017), 21–44; Izv. Math., 81:3 (2017), 481–504

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8478
• https://doi.org/10.4213/im8478
• http://mi.mathnet.ru/eng/izv/v81/i3/p21

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This publication is cited in the following articles:
1. V. Beloshapko, “Special properties of the stationary solution for two-dimensional singularly perturbed parabolic problem, stability, and attraction domain”, Math. Methods Appl. Sci., 41:18 (2018), 9264–9275
2. V. F. Butuzov, “Asymptotic behaviour of a boundary layer solution to a stationary partly dissipative system with a multiple root of the degenerate equation”, Sb. Math., 210:11 (2019), 1581–1608
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