# Weighted *L*^{2}-estimates of solutions for damped wave equations with variable coefficients

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## Abstract

The authors establish weighted *L*^{2}-estimates of solutions for the damped wave equations with variable coefficients *u*_{ tt }− divA(*x*)∇*u*+*au*_{ t } = 0 in *ℝ*^{ n } under the assumption a(*x*) ≥ a_{0}[1+*ρ*(*x*)]^{−l}, where a_{0} > 0, *l* < 1, *ρ*(*x*) is the distance function of the metric *g* = *A*^{−1}(*x*) on *ℝ*^{ n }. The authors show that these weighted *L*^{2}-estimates are closely related to the geometrical properties of the metric *g* = *A*^{−1}(*x*).

## Keywords

Distance function of a metric Riemannian metric wave equation with variable coefficients weighted*L*

^{2}-estimate

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