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.
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
\Bibitem{Kas19}
\by S.~A.~Kashchenko
\paper Homogenization over the spatial variable in nonlinear parabolic systems
\serial Tr. Mosk. Mat. Obs.
\yr 2019
\vol 80
\issue 1
\pages 63--86
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo618}
\elib{https://elibrary.ru/item.asp?id=43285881}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2019
\vol 80
\pages 53--71
\crossref{https://doi.org/10.1090/mosc/288}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85083777349}
Linking options:
https://www.mathnet.ru/eng/mmo618
https://www.mathnet.ru/eng/mmo/v80/i1/p63
This publication is cited in the following 3 articles:
I. V. Boikov, “Ustoichivost reshenii sistem parabolicheskikh uravnenii s zapazdyvaniyami”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 2023, no. 1, 69–84
I. V. Boykov, “Stability of Solutions to Systems of Parabolic Equations with Delay”, Tech. Phys., 68:3 (2023), 59
S. A. Kaschenko, D. O. Loginov, “The influence of external environment resistance coefficient on population dynamics”, Russian Math. (Iz. VUZ), 66:1 (2022), 53–61
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