Existence and Boundary Stabilization of a Nonlinear Hyperbolic Equation with Time-dependent Coefficients

Abstract

In this article, we study the hyperbolic problem

K(x,t)u{tt} - \∑n j=1 (a(x,t)uxj) + F(x,t,u,∇u) = 0

u = 0 on Γ1, ∂u/∂v + ß(x)ut = 0 on Γ0

u(0) = u0, ut(0) = u1 in Ω,

where Ω is a bounded region in Rn 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.