Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients

In this article, we study the hyperbolic problem $$ K(x,t)u_{tt} - sum_{j=1}^nleft(a(x,t)u_{x_j} ight) + F(x,t,u,abla u) = 0 $$ coupled with boundary conditions $$u=0,quadhbox{on }Gamma_1,, quad {partial u overpartialu} + Beta(x)u_t =0quadhbox{ on }Gamma_0,.$$ Here the variable $x$ belongs to a bounded region of ${Bbb R}^n$, whose boundary is partitioned into two disjoint sets $Gamma_0,Gamma_1$.