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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
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