## 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

### Keywords:

Riemannian metric; localized dissipation near infinity

### Citations:

Zbl 1002.35079; Zbl 0951.35069
Full Text:

### References:

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