Uniform stabilization of the quasi-linear Kirchhoff wave equation with a nonlinear boundary feedback

An n-dimensional quasi-linear wave equation defined on bounded domain Omega with Neumann boundary conditions imposed on the boundary Gamma and with a nonlinear boundary feedback acting on a portion of the boundary Gamma(1) su bset of Gamma is considered. Global existence, uniqueness and uniform decay rates are established for the model, under the assumption that the H-1(Ome ga) x L-2(Omega) norms of the initial data are sufficiently small. The resu lt presented in this paper extends these obtained recently in Lasiecka, and Ong (1999), where the Dirichlet boundary conditions al-e imposed on the un controlled portion of the boundary Gamma(0) = Gamma \ <(Gamma(1))over bar>, and the two portions of the boundary are assumed disjoint, i.e. <(Gamma(1) )over bar> boolean AND <(Gamma(0))over bar> = 0. The goal of this paper is to remove this restriction. This is achieved by considering the "pure" Neum ann problem subject to convexity assumption imposed on Gamma(0).