The authors establish weighted L ²-estimates of solutions for the damped wave equations with variable coeﬃcients u _tt−divA(x)∇u+a(x)u _t=0 in Rn under some assumptions on the damping a(x). The authors show that these weighted L ²-estimates are closely related to the geometrical properties of the metric determined by A(x). Some examples are given about applications of those L ²-estimates in obtaining stabilization properties of wave equations.

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