Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $$ u_{tt}-u_{xx}+(\alpha_0+\varphi(x))u=0,\qquad t>0,\quad-\infty<x<\infty, $$ 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. 
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     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},
     year = {1975},
     volume = {26},
     number = {3},
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     url = {http://geodesic.mathdoc.fr/item/SM_1975_26_3_a6/}
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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/

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[2] B. R. Vainberg, “Povedenie pri bolshikh vremenakh reshenii uravneniya Kleina-Gordona”, Trudy Mosk. matem. obschestva, XXX (1974), 139–158

[3] D. Thoe, “On the exponential decay of solutions of the wave equation”, J. Math. Analisis and Appl., 16:2 (1966), 333–346 | DOI | MR | Zbl

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