On the global solvability of solutions to a quasilinear wave equation with localized damping and source terms

We prove existence and uniform stability of strong solutions to a quasilinear wave equation with a locally distributed nonlinear dissipation with source term of power nonlinearity of the typeu″−M(∫Ω|∇u|2dx)Δu+a(x)g(u′)+f(u)=0,inΩ×]0,+∞[,u=0,onΓ×]0,+∞[,u(x,0)=u0(x),u′(x,0)=u1(x), inΩ.