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 [u_tt - Δu + ρ(x, u_t) + β(x, u) = f (x, t) εΩ x R⁺(R ] with the boundary condition u _∂Ω, where Ω is a bounded domain in R^N, ρ(x, ν) is a function like ρ(x, ν) = a(x)g(ν) with g′(ν) ≥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.