Global Smooth Solutions for Quasilinear Wave Equation with Locally Internal Damping

We study the existence of global smooth solutions for the quasilinear wave equations with internal locally damping when initial data are near a given equilibrium. Our interest is to study the effect of the damping region which guarantees the existence of global solutions. Our results show that the structure of the damping region depends on geometric properties of a Riemannian metric, given by the variable coefficients and the equilibrium of the system. Some geometrical conditions are presented to obtain the damping region.