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Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation. (English) Zbl 1190.35036

The appropriate energy \(E(t)\) of the stated problem is defined first. In case of constant coefficients , using the multiplier technique in the \(n\)-dimensional Euclidian space, a decay of \(E(t)\) has been proved by M. Nakao [Math. Z. 238, No. 4, 781–797 (2001; Zbl 1002.35079)]. The aim here is to get a similar estimation in case of variable coefficients. Under a special condition regarding the Riemann metric, one proves the existence of a unique solution to the stated problem and one obtains an estimation of Nakao type. The special condition has been introduced by P. F. Yao [SIAM J. Control Optim. 37, No. 5, 1568–1599 (1999; Zbl 0951.35069)].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
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