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Mat. Sb. (N.S.), 1981, Volume 115(157), Number 1(5), Pages 130–145 (Mi msb2378)  

This article is cited in 1 scientific paper (total in 1 paper)

On the limit behavior of the domain of dependence of a hyperbolic equation with rapidly oscillating coefficients

A. L. Piatnitski


Abstract: In this paper, the behavior of the support of the solution to the Cauchy problem for a hyperbolic equation of the form
$$ \frac{\partial^2}{\partial t^2}u^\varepsilon(x, t)-\frac\partial{\partial x_i}a_{ij}(\frac x\varepsilon)\frac\partial{\partial x_j}u^\varepsilon+b_i(x, \frac x\varepsilon)\frac\partial{\partial x_i}u^\varepsilon+c(x, \frac x\varepsilon)u^\varepsilon=0 $$
with periodic, rapidly oscillating coefficients $a_{ij}(y)$ and small parameter $\varepsilon$, is studied. It is proved that, for small $\varepsilon$, the domain of dependence of this equation is close to some convex cone with rectilinear generators.
In the case when the coefficients $a_{ij}$ depend essentially on only one argument, e.g. $y_1$, this limit cone can be found explicitly. Its construction uses the Hamiltonian, which does not depend on $\varepsilon$ and does not correspond to any differential operator.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1982, 43:1, 117–131

Bibliographic databases:

UDC: 517.946
MSC: 35L15, 35B20, 35B40
Received: 24.03.1980

Citation: A. L. Piatnitski, “On the limit behavior of the domain of dependence of a hyperbolic equation with rapidly oscillating coefficients”, Mat. Sb. (N.S.), 115(157):1(5) (1981), 130–145; Math. USSR-Sb., 43:1 (1982), 117–131

Citation in format AMSBIB
\Bibitem{Pia81}
\by A.~L.~Piatnitski
\paper On the limit behavior of the domain of dependence of a~hyperbolic equation with rapidly oscillating coefficients
\jour Mat. Sb. (N.S.)
\yr 1981
\vol 115(157)
\issue 1(5)
\pages 130--145
\mathnet{http://mi.mathnet.ru/msb2378}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=618591}
\zmath{https://zbmath.org/?q=an:0494.35014|0459.35012}
\transl
\jour Math. USSR-Sb.
\yr 1982
\vol 43
\issue 1
\pages 117--131
\crossref{https://doi.org/10.1070/SM1982v043n01ABEH002435}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Lions J., “Exact Controllability, Stabilization and Perturbations for Distributed Systems”, SIAM Rev., 30:1 (1988), 1–68  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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