Precise decay rate estimates for time-dependent dissipative systems

We consider the wave equation damped with a nonlinear time-dependent distributed dissipation. By generalizing a method recently introduced to study autonomous systems, we show that the energy of the system decays to zero with an explicit and precise decay rate estimate under sharp assumptions on the feedback. Then we prove that our estimates are optimal for the problem of the one dimensional wave equation damped by a nonlinear time-dependent boundary feedback. This extends and improves several earlier results of E. Zuazua and M. Nakao, and completes strong stability results of P. Pucci and J. Serrin.