Global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation

We consider the existence of global solutions of the nonlinear elastodynamic system with a locally distributed damping in a bounded domain. We assume that the energy function satisfies the strong ellipticity condition at the zero equilibrium. The local dissipation is in the form a(x)u̇ where the nonnegative function a(x)a(x) is only positive on a small portion of the domain. We show the existence of global smooth solutions when initial data are small. In particular, we obtain the exponential decay of the energy, which implies the exponential stabilization of the system by internal feedback.