Decay of solutions of the wave equation with a local highly degenerate dissipation

We consider the system of the wave equation with the Dirichlet boundary condition damped with a local linear dissipation of the type $a(x)u'$ . In order to simplify, we assume that the domain is the open ball of RN centered in O and of radius R and that the function a is radial near the boundary. When a goes quickly to zero at the boundary, we obtain a precise decay rate estimate of the energy of the solutions. This extends some results of Zuazua and Nakao. The proof is based on a new nonlinear integral inequality and on the asymptotic behavior of the function a at the boundary.