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 Mat. Sb., 1999, Volume 190, Number 9, Pages 99–126 (Mi msb427)

Homogenization of attractors of non-linear hyperbolic equations with asymptotically degenerate coefficients

L. S. Pankratova, I. D. Chueshovb

a B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine
b V. N. Karazin Kharkiv National University

Abstract: A non-linear initial-boundary-value problem for a hyperbolic equation with dissipation is considered in a bounded domain $\Omega$
$$u^\varepsilon _{tt}+\delta u^\varepsilon _t -\operatorname {div}(a^\varepsilon (x)\nabla u^\varepsilon) +f(u^\varepsilon)=h^\varepsilon (x),$$
where $\delta>0$ and the coefficient $a^\varepsilon (x)$ is of order $\varepsilon ^{3+\gamma}$ $(0\leqslant \gamma<1)$ on the union of spherical annuli of thickness $d_\varepsilon=d\varepsilon^{2+\gamma}$. The annuli are periodically, with period $\varepsilon$, distributed in a bounded domain $\Omega$. Outside the union of the annuli $a^\varepsilon (x)\equiv 1$. The asymptotic behaviour of the solutions and the global attractor of the problem are studied as $\varepsilon \to 0$. It is shown that the homogenization of the problem on each finite time interval leads to a system consisting of a non-linear hyperbolic equation and an ordinary second-order differential equation (with respect to $t$). It is also shown that the global attractor of the initial problem approaches in a certain sense a weak global attractor of the homogenized problem.

DOI: https://doi.org/10.4213/sm427

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English version:
Sbornik: Mathematics, 1999, 190:9, 1325–1352

Bibliographic databases:

UDC: 517.953
MSC: 35B27, 35B40, 35L70

Citation: L. S. Pankratov, I. D. Chueshov, “Homogenization of attractors of non-linear hyperbolic equations with asymptotically degenerate coefficients”, Mat. Sb., 190:9 (1999), 99–126; Sb. Math., 190:9 (1999), 1325–1352

Citation in format AMSBIB
\Bibitem{PanChu99} \by L.~S.~Pankratov, I.~D.~Chueshov \paper Homogenization of attractors of non-linear hyperbolic equations with asymptotically degenerate coefficients \jour Mat. Sb. \yr 1999 \vol 190 \issue 9 \pages 99--126 \mathnet{http://mi.mathnet.ru/msb427} \crossref{https://doi.org/10.4213/sm427} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1725227} \zmath{https://zbmath.org/?q=an:0940.35029} \transl \jour Sb. Math. \yr 1999 \vol 190 \issue 9 \pages 1325--1352 \crossref{https://doi.org/10.1070/sm1999v190n09ABEH000427} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000085043300004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0033236627} 

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• https://doi.org/10.4213/sm427
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