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On uniform quasistabilization of solutions of the second mixed problem for a hyperbolic equation. (English. Russian original) Zbl 0617.35070
Sov. Math., Dokl. 33, 334-337 (1986); translation from Dokl. Akad. Nauk SSSR 287, 45-49 (1986).
The problem examined in this work is the quasistabilization of solutions of the following problem \[ \partial^ 2u/\partial t^ 2=\sum^{n}_{i,j=1}(\partial /\partial x_ i)(a_{ij}(x)\partial u/\partial x_ j),\quad x\in \Omega,\quad t>0 \] \[ (\partial u/\partial N)|_{x\in \partial \Omega,t>0}=0,\quad u|_{x\in \Omega,t=0}=\phi (x),\quad (\partial u/\partial t)|_{x\in \Omega,t=0}=0, \] where the symmetric matrix \(A(x)=a_{ij}(x)\) satisfies the condition \(\gamma^{-1}| \xi |^ 2 \leq (\xi,A(x)\xi)\leq \gamma | \xi |^ 2\), xE\(\Omega\), \(\xi \in R^ n\), \(\gamma >0\), \(n>1\) and \(\Omega \subset R^ n\) is an unbounded domain.
The criterion of uniform quasistabilization of the solution of this problem for certain class of \(\Omega\) is formulated in the form of a theorem. The theorem is proved by means of a criterion for uniform stabilization of the solution of the second mixed problem for the related parabolic equation.
Reviewer: G.Moluarka
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs