On~the behaviour for large values of the time of the solution of the Cauchy problem for the equation $\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}+\alpha(x)u=0$
Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 403-426
Voir la notice de l'article provenant de la source Math-Net.Ru
We obtain an asymptotic expansion as $t\to\infty$ for the solution $u(t,x)$ of the Cauchy problem with initial functions of compact support for the equation
$$
u_{tt}-u_{xx}+(\alpha_0+\varphi(x))u=0,\qquad t>0,\quad-\infty\infty,
$$
where $\alpha_0=\text{const}$ and $\varphi(x)$ satisfies the following condition for some $k\geqslant1$:
$$
\int_{-\infty}^\infty|x|^k|\varphi(x)|\,dx\infty.
$$ Bibliography: 4 titles.
@article{SM_1975_26_3_a6,
author = {S. A. Laptev},
title = {On~the behaviour for large values of the time of the solution of the {Cauchy} problem for the equation $\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}+\alpha(x)u=0$},
journal = {Sbornik. Mathematics},
pages = {403--426},
publisher = {mathdoc},
volume = {26},
number = {3},
year = {1975},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1975_26_3_a6/}
}
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JO - Sbornik. Mathematics
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%J Sbornik. Mathematics
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S. A. Laptev. On~the behaviour for large values of the time of the solution of the Cauchy problem for the equation $\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}+\alpha(x)u=0$. Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 403-426. http://geodesic.mathdoc.fr/item/SM_1975_26_3_a6/