Stabilization of the nonlinear elastodynamic system with a boundary dissipation

We consider the existence of global solutions of the nonlinear elastodynamic system with a boundary dissipation structure when initial data are small. The medium material is assumed to be isotropic and hyperelastic. We show the existence of global smooth solutions. In particular, we obtain the exponential decay of the energy, which implies the exponential stabilization of the system by boundary feedback.