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 Ω with Neumann boundary conditions imposed on the boundary Γ and with a nonlinear boundary feedback acting on a portion of the boundary Γ1⊂ ⊂ Γ is considered. Global existence, uniqueness and uniform decay rates are established for the model, under the assumption that the H1(Ω) × L2(Ω) norms of the initial data are sufficiently small. The result presented in this paper extends these obtained recently in Lasiecka and Ong (1999), where the Dirichlet boundary conditions are imposed on the uncontrolled portion of the boundary Γ0 = Γ \ Γ1̄, and the two portions of the boundary are assumed disjoint, i.e. Γ1̄∩Γ0 = θ. The goal of this paper is to remove this restriction. This is achieved by considering the pure Neumann problem subject to convexity assumption imposed on Γ0