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

This article is cited in 14 scientific papers (total in 14 papers)

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.


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

Bibliographic databases:

UDC: 517.953
MSC: 35B27, 35B40, 35L70
Received: 05.10.1998

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
\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
\jour Sb. Math.
\yr 1999
\vol 190
\issue 9
\pages 1325--1352

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    This publication is cited in the following articles:
    1. Efendiev, M, “Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization”, Annales de l Institut Henri Poincare-Analyse Non Lineaire, 19:6 (2002), 961  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    2. A. M. Rekalo, I. D. Chueshov, “Global attractor of a contact parabolic problem in a thin two-layer domain”, Sb. Math., 195:1 (2004), 97–119  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Pankratov L., Chueshov I., “Non-linear acoustic oscillations in a strongly inhomogeneous medium”, 10Th International Conference on Mathematical Methods in Electromagnetic Theory, Conference Proceedings, 2004, 41–44  crossref  isi
    4. Zelik, S, “Global averaging and parametric resonances in damped semilinear wave equations”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 136 (2006), 1053  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    5. Cavalcanti, MM, “Homogenization for a nonlinear wave equation in domains with holes of small capacity”, Discrete and Continuous Dynamical Systems, 16:4 (2006), 721  crossref  mathscinet  zmath  isi  elib
    6. Chepyzhov, VV, “Averaging of nonautonomous damped wave equations with singularly oscillating external forces”, Journal de Mathematiques Pures et Appliquees, 90:5 (2008), 469  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. Jean Louis Woukeng, David Dongo, “Multiscale homogenization of nonlinear hyperbolic equations with several time scales”, Acta Mathematica Scientia, 31:3 (2011), 843  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    8. Khrabustovskyi A., “Periodic Elliptic Operators with Asymptotically Preassigned Spectrum”, Asymptotic Anal., 82:1-2 (2013), 1–37  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    9. Khrabustovskyi A., Plum M., “Spectral properties of an elliptic operator with double-contrast coefficients near a hyperplane”, Asymptotic Anal., 98:1-2 (2016), 91–130  crossref  mathscinet  zmath  isi  scopus
    10. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V. Goritsky A.Yu., “Homogenization of trajectory attractors of 3D Navier–Stokes system with randomly oscillating force”, Discret. Contin. Dyn. Syst., 37:5 (2017), 2375–2393  crossref  mathscinet  zmath  isi  scopus
    11. Chechkin G.A. Chepyzhov V.V. Pankratov L.S., “Homogenization of Trajectory Attractors of Ginzburg-Landau Equations With Randomly Oscillating Terms”, Discrete Contin. Dyn. Syst.-Ser. B, 23:3 (2018), 1133–1154  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    12. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V., “Weak Convergence of Attractors of Reaction-Diffusion Systems With Randomly Oscillating Coefficients”, Appl. Anal., 98:1-2, SI (2019), 256–271  crossref  mathscinet  zmath  isi  scopus
    13. Cooper Sh., Savostianov A., “Homogenisation With Error Estimates of Attractors For Damped Semi-Linear Anisotropic Wave Equations”, Adv. Nonlinear Anal., 9:1 (2020), 745–787  crossref  mathscinet  zmath  isi
    14. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V., “Strong Convergence of Trajectory Attractors For Reaction-Diffusion Systems With Random Rapidly Oscillating Terms”, Commun. Pure Appl. Anal, 19:5 (2020), 2419–2443  crossref  mathscinet  zmath  isi
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