Higher Order Energy Decay Rates for Damped Wave Equations with Variable Coefficients

Abstract. Under appropriate assumptions the energy of wave equations with damping and variable coefficients c(x)utt − div(b(x)∇u) + a(x)ut = h(x, t) has been shown to decay. Determining the decay rate for the higher order energies of the kth order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for k = 1, 2. We establish the sharp gain in the decay rate for all higher order energies in terms of the first energy, and also obtain the sharp gain of decay rates for the L norms of the higher order spatial derivatives. The results concern weighted (in time) and also pointwise (in time) energy decay estimates. We also obtain L∞ estimates for the solution u in dimension n = 3. As an application we compute explicit decay rates for all energies which involve the dimension n and the bounds for the coefficients a(x) and b(x) in the case c(x) = 1 and h(x, t) = 0.