RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (1531 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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} 

• http://mi.mathnet.ru/eng/msb2378
• http://mi.mathnet.ru/eng/msb/v157/i1/p130

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 194 Full text: 57 References: 36 First page: 1