Polynomial decay rate for the dissipative wave equation

This paper is devoted to study the stabilization of the linear wave equation in a bounded domain damped in a subdomain when the geometrical control condition (see [ BLR]) of the work of C. Bardos, G. Lebeau and J. Rauch is not fulfilled. In such case, they [ BLR] proved that the uniform exponential decay rate of the energy cannot be hoped due to the existence of a trapped ray that never reaches the support of the damping. Another important contribution in this field was done by G. Lebeau [ Le] who establishes a logarithmic decay rate for the dissipative wave equation when no assumption on rays of geometrical optics is required, but when more regularity on the initial data is allowed. Now, it seems natural to search a general description of the geometries of both domain and support of the damping under which the energy of the dissipative wave equation decreases in a polynomial way. A first answer in this direction was done by Z. Liu and B. Rao [ LR] who consider the wave equation on a square damped in a vertical strip. In this paper, we improve the geometry to a partially cubic domain where the damping acts in a neighborhood of the boundary except between a pair of parallel square face of the cube.