Decay property for second order hyperbolic systems of viscoelastic materials

We study a class of second order hyperbolic systems with dissipation which describes viscoelastic materials. The considered dissipation is given by the sum of the memory term and the damping term. When the dissipation is effective over the whole system, we show that the solution decays in L² at the rate t^{- n / 4} as t → ∞, provided that the corresponding initial data are in L² ∩ L¹, where n is the space dimension. The proof is based on the energy method in the Fourier space. Also, we discuss similar systems with weaker dissipation by introducing the operator (1 - Δ)^{- θ / 2} with θ > 0 in front of the dissipation terms and observe that the decay structure of these systems is of the regularity-loss type.