Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation

We discuss the existence of global or periodic solutions to the nonlinear wave equation [utt−Δu+ρ(x,ut)+β(x,u)=f(x,t) Ω×R+(R)] with the boundary condition u∂Ω, where Ω is a bounded domain in RN,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v) 0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting