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$%P. Martínez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut. 12 (1999), no. 1, 251283.%$

A New Method to Obtain Decay Rate Estimates for Dissipative Systems with Localized Damping
Patrick MARTÍNEZ
Institut de Recherche Mathématique Avancée
Université Louis Pasteur, 7
rue René Descartes
67084 Strasbeurg Cédex — France

Received: April 15, 1998
Revised: May 27, 1998

ABSTRACT

We consider the wave equation damped with a locally distributed nonlinear dissipation. We improve several earlier results of E. Zuazua and of M. Nakao in two directions: first, using the piecewise multiplier method introduced by K. Liu, we weaken the usual geometrical conditions on the localization of the damping. Then thanks to some new nonlinear integral inequalities, we eliminate the usual assumption on the polynomial growth of the feedback in zero and we show that the energy of the system decays to zero with a precise decay rate estimate.

1991 Mathematics Subject Classification: 26A12, 25B40, 93D15.