Stabilization and Control for the Subcritical Semilinear Wave Equation in a Bounded Domain With a Cauchy-Ventcel Boundary Conditions

The exponential decay property of solutions of the semilinear wave equation in bounded domain of R^N(N is equals or greater than 1) with a damping term which is effective on the exterior of a ball and with boundary conditions of Cauchy-Ventcel type was analyzed. Under suitable and natural assumptions on the nonlinearity, it was proved that the exponential decay holds locally uniformly for finite energy solutions that provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity atmost as a power is less than 5. The results obtained in R³ and R^N (N equals to or greater than 1) by B. Dehman, G. Le beau 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 contro llability of the semilinear wave equation in abounded domain of R^N (N equals to orgreater than 1) with a subcritical nonlinearity on the domain and its boundary and with conditions on the boundary of Cauchy-Ventcel type.