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Sbornik: Mathematics, 2003, Volume 194, Issue 9, Pages 1273–1300
DOI: https://doi.org/10.1070/SM2003v194n09ABEH000765
(Mi sm765)
 

This article is cited in 19 scientific papers (total in 19 papers)

Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time

M. I. Vishik, V. V. Chepyzhov

Institute for Information Transmission Problems, Russian Academy of Sciences
References:
Abstract: A quasilinear dissipative wave equation is considered for periodic boundary conditions with exterior force $g(x,t/\varepsilon)$ rapidly oscillating in $t$. It is assumed in addition that, as $\varepsilon\to0+$, the function $g(x,t/\varepsilon)$ converges in the weak sense (in $L_{2,w}^{\mathrm{loc}}(\mathbb R,L_2(\mathbb T^n))$ to a function $\overline g(x)$ and the averaged wave equation (with exterior force $\overline g(x)$ has only finitely many stationary points $\{z_i(x),\,i= 1,\dots,N\}$, each of them hyperbolic. It is proved that the global attractor $\mathscr A_\varepsilon$ of the original equation deviates in the energy norm from the global attractor $\mathscr A_0$ of the averaged equation by a quantity $C\varepsilon^\rho$, where $\rho$ is described by an explicit formula. It is also shown that each piece of a trajectory $u^\varepsilon(t)$ of the original equation lying on $\mathscr A_\varepsilon$ that corresponds to an interval of time-length $C\log(1/\varepsilon)$ can be approximated to within $C_1\varepsilon^{\rho_1}$ by means of finitely many pieces of trajectories lying on unstable manifolds $M^u(z_i)$ of the averaged equation, where an explicit expression for $\rho_1$ is provided.
Received: 21.03.2003
Russian version:
Matematicheskii Sbornik, 2003, Volume 194, Number 9, Pages 3–30
DOI: https://doi.org/10.4213/sm765
Bibliographic databases:
UDC: 517.9
MSC: Primary 35B41, 34C29; Secondary 35L70
Language: English
Original paper language: Russian
Citation: M. I. Vishik, V. V. Chepyzhov, “Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time”, Mat. Sb., 194:9 (2003), 3–30; Sb. Math., 194:9 (2003), 1273–1300
Citation in format AMSBIB
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\by M.~I.~Vishik, V.~V.~Chepyzhov
\paper Approximation of trajectories lying on a~global attractor of a~hyperbolic equation with exterior force rapidly oscillating in time
\jour Mat. Sb.
\yr 2003
\vol 194
\issue 9
\pages 3--30
\mathnet{http://mi.mathnet.ru/sm765}
\crossref{https://doi.org/10.4213/sm765}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2037501}
\zmath{https://zbmath.org/?q=an:1077.37048}
\transl
\jour Sb. Math.
\yr 2003
\vol 194
\issue 9
\pages 1273--1300
\crossref{https://doi.org/10.1070/SM2003v194n09ABEH000765}
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  • This publication is cited in the following 19 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Russian version PDF:217
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    References:71
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