On a point source in an inhomogeneous medium

B. R. Vainberg

Sbornik. Mathematics, Tome 23 (1974) no. 1, pp. 123-148

Résumé

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.

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