Decay rate estimates for a weak viscoelastic beam equation with time-varying delay

We consider a weak viscoelastic beam equation with internal time-varying delay utt+Δ2u−M(‖∇u‖2)Δu+σ(t)∫0tg(t−s)Δu(s)ds+0ut+a1ut(x,t−τ(t))=0. By introducing suitable energy and Lyapunov functionals, we establish general decay estimates for the energy, which depends on the behavior of both σσ and gg.