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Mathematics of the USSR-Sbornik, 1974, Volume 23, Issue 1, Pages 123–148
DOI: https://doi.org/10.1070/SM1974v023n01ABEH001716
(Mi sm3661)
 

This article is cited in 4 scientific papers (total in 4 papers)

On a point source in an inhomogeneous medium

B. R. Vainberg
References:
Abstract: Let $L\bigl(x,\frac\partial{\partial x}\bigr)$, $x\in\mathbf R^n$, be a second-order elliptic differential operator coinciding with the Laplace operator in a neighborhood of infinity. Let $E$ be the Green's function of the Cauchy problem for the operator $\frac{\partial^2}{\partial t^2}-L$. Under certain assumptions regarding the trajectories of the Hamiltonian system connected with the operator in question, the following results are obtained: 1) an asymptotic approximation with respect to smoothness $E_N$ to the function $E$ is constructed by Hadamard's method; 2) we show that the Fourier transformation of $E_N$ from $t$ to $k$ is an analytic function of $k$ in the complex plane with a cut along the negative part of the imaginary axis, and with $\lvert\operatorname{Im}k\rvert<C<\infty$ and $\lvert\operatorname{Re}k\rvert\to\infty$ it gives the asymptotic behavior of the fundamental solution of the operator $-L-k^2$; 3) the asymptotic behavior as $t\to\infty$ of the solutions of the nonstationary problem is obtained.
Bibliography: 44 titles.
Received: 26.06.1973
Bibliographic databases:
UDC: 517.944
MSC: Primary 35L15, 35B40, 35A35; Secondary 35A22, 35P25
Language: English
Original paper language: Russian
Citation: B. R. Vainberg, “On a point source in an inhomogeneous medium”, Math. USSR-Sb., 23:1 (1974), 123–148
Citation in format AMSBIB
\Bibitem{Vai74}
\by B.~R.~Vainberg
\paper On a~point source in an inhomogeneous medium
\jour Math. USSR-Sb.
\yr 1974
\vol 23
\issue 1
\pages 123--148
\mathnet{http://mi.mathnet.ru//eng/sm3661}
\crossref{https://doi.org/10.1070/SM1974v023n01ABEH001716}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=342864}
\zmath{https://zbmath.org/?q=an:0293.35046}
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  • https://doi.org/10.1070/SM1974v023n01ABEH001716
  • https://www.mathnet.ru/eng/sm/v136/i1/p126
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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