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Trudy Moskovskogo Matematicheskogo Obshchestva, 2019, Volume 80, Issue 1, Pages 63–86
(Mi mmo618)
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This article is cited in 3 scientific papers (total in 3 papers)
Homogenization over the spatial variable in nonlinear parabolic systems
S. A. Kashchenkoab a P. G. Demidov Yaroslavl State University, Yaroslavl, 150003 Russia
b National Research Nuclear University MEPhI, Moscow, 115409 Russia
Abstract:
We consider boundary value problems for nonlinear parabolic systems whose coefficients are periodic rapidly oscillating functions of the spatial variable. Results on the closeness of time-periodic solutions of an original boundary value problem and the problem homogenized over the spatial variable are presented. The dynamic properties of these equations are studied in near-critical cases of the equilibrium stability problem. Algorithms for constructing the asymptotics of periodic solutions and for calculating the coefficients of the so-called normal forms are developed. In particular, we show that an infinite process of bifurcation and disappearance of a stable cycle can occur with increasing oscillation degree of the coefficients. In addition, we study some classes of problems with a deviation in the spatial variable as well as with a large diffusion coefficient. Logistic delay equations with diffusion and logistic equations with a deviation in the spatial variable, which are important in applications, are studied as examples. The coefficients of these equations are assumed to be rapidly oscillating in the spatial variable.
Key words and phrases:
Nonlinear parabolic system, boundary value problem, rapidly oscillating data, stability, bifurcation.
Received: 14.05.2018
Citation:
S. A. Kashchenko, “Homogenization over the spatial variable in nonlinear parabolic systems”, Tr. Mosk. Mat. Obs., 80, no. 1, MCCME, M., 2019, 63–86; Trans. Moscow Math. Soc., 80 (2019), 53–71
Linking options:
https://www.mathnet.ru/eng/mmo618 https://www.mathnet.ru/eng/mmo/v80/i1/p63
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Abstract page: | 164 | Full-text PDF : | 57 | References: | 19 | First page: | 2 |
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