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

In this article, we study the hyperbolic problem K(x, t)utt − ∑n j=1 ( a(x, t)uxj ) + F (x, t, u,∇u) = 0 u = 0 on Γ1, ∂u ∂ν + β(x)ut = 0 on Γ0 u(0) = u, ut(0) = u 1 in Ω , where Ω is a bounded region in R whose boundary is partitioned into two disjoint sets Γ0,Γ1. We prove existence, uniqueness, and uniform stability of strong and weak solutions when the coefficients and the boundary conditions provide a damping effect.