Polynomial decay rate for the dissipative wave equation

We study the dissipative linear wave equation in a bounded domain. The exponential decay rate of the energy was established by Bardos, Lebeau and Rauch under a geometrical hypothesis linked with the geodesics. Furthermore such condition called geometric control condition is almost necessary to get a uniform exponential decay. In another hand, Lebeau proved a logarithmic decay rate for smooth solutions when no particular geometric condition is required. In this paper we give for some particular geometries a polynomial decay rate when the geometric control condition is not fulfilled.