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On uniform quasistabilization of solutions of the second mixed problem for a second-order hyperbolic equation. (English. Russian original) Zbl 0628.35055
Math. USSR, Sb. 57, 243-262 (1987); translation from Mat. Sb., Nov. Ser. 129(171), No. 2, 232-251 (1986).
Using some results by A. K. Gushchin, V. P. Mikhajlov and the author [ibid. 56, 141-162 (1987); translation from Mat. Sb., Nov. Ser. 128(170), 147-168 (1985)], the author proves, that under some conditions on the region $$\Omega \subset R^ n$$ and $$\phi$$ the average $$(\alpha /t^{\alpha})\int^{t}_{0}(t-s)^{\alpha} u(x,s) ds$$ of order $$\alpha >[n/2]+1$$ of a solution u(x,t) of the equation $$u_{tt}=\sum^{n}_{i,j=1}(a_{ij}(x)u_{x_ j})_{x_ i}$$ with Neumann boundary conditions and the initial data $$u(x,0)=\phi (x)$$, $$u_ t(x,0)=0$$ tends (uniformly with respect to x) to zero for $$t\to \infty$$ if and only if the average $(mess K(x,R)\cap \Omega)^{- 1}\int_{K(x,R)\cap \Omega}\phi (y) dy\quad (R\to \infty)$ tends to zero. The paper follows similar results for a Cauchy problem by A. K. Gushchin and V. P. Mikhajlov [Tr. Mat. Inst. Steklova 166, 76- 90 (1984; Zbl 0567.35052)].
Reviewer: M.Kopáčková
##### MSC:
 35L20 Initial-boundary value problems for second-order hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs
##### Keywords:
Neumann boundary conditions; initial data; average
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