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 u t t + Δ 2 u − M ( ‖ ∇ u ‖ 2 ) Δ u + σ ( t ) ∫ 0 t g ( t − s ) Δ u ( s ) d s + 0 u t + a 1 u t ( 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 σ and g .