Existence and boundary stabilization of the semilinear wave equation

We consider the semilinear wave equation u″−μ(t)Δu+h(u)=f in a bounded domain Q⊂Rn+1Q⊂Rn+1 with smooth boundary ΣΣ subject to mixed boundary conditions u=0u=0 on Σ0Σ0 and μ(∂u/∂ν)+βu′=0μ(∂u/∂ν)+βu′=0 on Σ1Σ1, {Σ0,Σ1}{Σ0,Σ1} being a partition on ΣΣ suitably chosen. We investigate existence, uniqueness and asymptotic behavior (as t→∞t→∞) of solution considering hh a real continuous function satisfying a sign assumption and ff, ββ and μμ under some adequate conditions.