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 Mat. Sb. (N.S.), 1974, Volume 94(136), Number 1(5), Pages 126–151 (Mi msb3661)

On a point source in an inhomogeneous medium

B. R. Vainberg

Abstract: Let $L(x,\frac\partial{\partial x})$, $x\in\mathbf R^n$, 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.

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English version:
Mathematics of the USSR-Sbornik, 1974, 23:1, 123–148

Bibliographic databases:

UDC: 517.944
MSC: Primary 35L15, 35B40, 35A35; Secondary 35A22, 35P25

Citation: B. R. Vainberg, “On a point source in an inhomogeneous medium”, Mat. Sb. (N.S.), 94(136):1(5) (1974), 126–151; 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 Mat. Sb. (N.S.) \yr 1974 \vol 94(136) \issue 1(5) \pages 126--151 \mathnet{http://mi.mathnet.ru/msb3661} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=342864} \zmath{https://zbmath.org/?q=an:0293.35046} \transl \jour Math. USSR-Sb. \yr 1974 \vol 23 \issue 1 \pages 123--148 \crossref{https://doi.org/10.1070/SM1974v023n01ABEH001716} 

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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. B. R. Vainberg, “On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as $t\to\infty$ of solutions of non-stationary problems”, Russian Math. Surveys, 30:2 (1975), 1–58
2. Pierre H. Bérard, “On the wave equation on a compact Riemannian manifold without conjugate points”, Math Z, 155:3 (1977), 249
3. A. B. Bakushinskii, A. S. Leonov, “Low-cost numerical method for solving a coefficient inverse problem for the wave equation in three-dimensional space”, Comput. Math. Math. Phys., 58:4 (2018), 548–561
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