Time decay of solutions of semilinear strongly damped generalized wave equations

We study the asymptotic behaviour in time of the solutions of dissipative perturbations of wave-type equations in ?^N, u_tt + Bu_t + Au + G(u) = 0, with commuting positive operators A, B and a power like non-linearity G(u). First we give some (pseudo) conformal invariants of the linear operator in the equation. This allows us to derive optimal decay rates for the solutions of the linearized problems. We then prove some decay estimates for the non-linear problems using the tools of scattering theory and the aforementioned conformal invariants.