Weighted [formula omitted]-estimates for dissipative wave equations with variable coefficients

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We establish weighted L 2 -estimates for the wave equation with variable damping u t t − Δ u + a u t = 0 in R n , where a ( x ) ⩾ a 0 ( 1 + | x | ) − α with a 0 > 0 and α ∈ [ 0 , 1 ) . In particular, we show that the energy of solutions decays at a polynomial rate t − ( n − α ) / ( 2 − α ) − 1 if a ( x ) ∼ a 0 | x | − α for large | x | . We derive these results by strengthening significantly the multiplier method. This approach can be adapted to other hyperbolic equations with damping.