Higher order energy decay for damped wave equations with variable coefficients

Under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c ( x ) u t t − div ( A ( x ) ∇ u ) + a ( x ) u t = 0 in R n are established. The results concern weighted (in time) and pointwise (in time) energy decay estimates. We also obtain weighted L 2 estimates for spatial derivatives.