Weighted L2-estimates for dissipative wave equations with variable coefficients

We establish weighted L2-estimates for the wave equation with variable damping utt−Δu+aut=0 in Rn, where a(x)⩾a0(1+|x|)−α with a0>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)∼a0|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.