Stabilization of the wave equation with variable coefficients and boundary condition of memory type

We consider the stabilization of the wave equation with space variable coefficients in a bounded region with a smooth boundary, subject to Dirichlet boundary conditions on one part of the boundary and linear or nonlinear dissipative boundary conditions of memory type on the remainder part of the boundary. Our stabilization results are mainly based on the use of differential geometry arguments, on the multiplier method and the introduction of suitable Lyapounov functionals.