Uniform stabilization for the semilinear wave equation in an inhomogeneous medium with locally distributed nonlinear damping and dynamic Cauchy–Ventcel type boundary conditions

In this paper, we consider the Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a nonlinear damping distributed around a neighborhood [Formula: see text] of the boundary according to the Geometric Control Condition. Uniform decay rates of the associated energy are established and, in addition, the exact internal controllability for the linear problem is also proved. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [Contrôle Optimal des équations aux dérivées partielles. (2001); http://www.math.upsud.fr/burq/articles/coursX.pdf ].