A new method to obtain decay rate estimates for dissipative systems with localized damping

Wc corasider tite wave equation dairaped with a localhy distributed nonlinear diasipation. Wc impnove several earlier results of E. Zuazua and of M. Nakan in two directiens: flrst, using the piecewse multiplier rnethod introduced hy K. Liu, we weaken the usual geornetrical coraditiens en thxe localizatien of the damping. Tben thanks te some new nonlinear integral inequalities, Wc eliminate the usual assumption on dic polynomial growth of the feedback mo zere arad we show that dic energy of the system decays to zero with a precise decay rate estimate.