Geodesic
On uniform quasistabilization of solutions of the second mixed problem for a second-order hyperbolic equation
Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 243-262 
Yu. A. Mikhailov. On uniform quasistabilization of solutions of the second mixed problem for a second-order hyperbolic equation. Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 243-262. http://geodesic.mathdoc.fr/item/SM_1987_57_1_a15/
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     title = {On uniform quasistabilization of solutions of the second mixed problem for a~second-order hyperbolic equation},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The problem \begin{gather*} u_{tt}(x,t)=\operatorname{div}_x(A(x)\nabla_xu(x,t)),\qquad x\in\Omega,\quad t>0; \\ \frac{\partial u}{\partial N}\bigg|_{\partial\Omega}=0;\quad u|_{t=0}=\varphi(x);\quad u_t|_{t=0}=0 \end{gather*} is considered in the cylindrical region $\Omega\times(0,+\infty)$. A criterion for uniform stabilization (with respect to $x$ in $\Omega$) of the mean over $t$ of order $\alpha$, $\alpha>[n/2]+1$, of the solution $u(x,t)$ of this problem is proved for a rather broad class of unbounded domains $\Omega\subset\mathbf R^n$ (determined by conditions of isoperimetric type). Bibliography: 15 titles.

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[3] Guschin A. K., Mikhailov V. P., Muravei L. A., “O stabilizatsii reshenii nestatsionarnykh granichnykh zadach dlya lineinykh differentsialnykh uravnenii v chastnykh proizvodnykh”, Dinamika sploshnoi sredy, 23 (1975), 57–91

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[6] Favorin V. M., “O povedenii pri $t\to\infty$ reshenii smeshannykh zadach dlya trekhmernogo volnovogo uravneniya v tsilindricheskoi po prostranstvennym peremennym oblasti”, Differents. uravneniya, 17:2 (1981), 354–365 | MR | Zbl

[7] Guschin A. K., Mikhailov V. P., “O ravnomernoi stabilizatsii resheniya zadachi Koshi dlya giperbolicheskogo uravneniya vtorogo poryadka”, Tr. MIAN, 166 (1984), 76–90 | Zbl

[8] Guschin A. K., Mikhailov V. P., Mikhailov Yu. A., “O ravnomernoi stabilizatsii resheniya vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 128(170) (1985), 147–168 | MR | Zbl

[9] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983 | MR

[10] Guschin A. K., “Ob otsenkakh reshenii kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka”, Tr. MIAN, 126 (1973), 5–45 | Zbl

[11] Khardi G., Littlvud Dzh., Polia G., Neravenstva, IL, M., 1948

[12] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[13] Guschin A. K., “O ravnomernoi stabilizatsii reshenii vtoroi smeshannoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 119(161) (1982), 451–508

[14] Wiener N., “Tauberian theorems”, Ann. of Math., 33 (1932), 1–100 | DOI | MR | Zbl

[15] Khardi G., Raskhodyaschiesya ryady, IL, M., 1951