Asymptotic stability of a viscoelastic problem with Balakrishnan-Taylor damping and time-varying delay

A viscoelastic problem with Balakrishnan-Taylordamping and time-varying delay of the form utt−(a+b‖∇u‖2+σ(∇u,∇ut))Δu+∫0tg(t−s)Δu(s)ds+μ1f1(ut(x,t))+μ2f2(ut(x,t−τ(t)))=0is considered. We prove a general stability result for the equation without the condition μ2>0 by establishing some Lyapunov functionals which are equivalent to the energy of the equation instead of multiplier technique and using some properties of convex functions.