Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions

We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domainΩof R~N with a damping term which is effective on the exterior of a ball and boundary conditions of the Canchy-Ventcel type.Under suitable and natural assumptions on the nonlinearity,we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means,roughly speaking,that the nonlinearity grows at infinity at most as a power p<5.The results obtained in R~3 and R~N by B.Dehman,G.Lebeau and E.Zuazua on the inequalities of the classical energy (which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball) and on Strichartz’s estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R~N with a subcritical nonlinearity on the domain and its boundary,and conditions on the boundary of Cauchy-Ventcel type.