Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping

In this paper we obtain an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation utt −∆u+ f(x, t, u) + ∫ t 0 g(t− τ)∆u(τ) dτ + a(x)ut = 0 in Ω× (0,∞). Here the damping term a(x)ut may be null for some part of the domain Ω. By assuming that the kernel g in the memory term decays exponentially, the damping effect allows us to avoid compactness arguments and and to reduce number of the energy estimates considered in the prior literature. We construct a suitable Liapunov functional and make use of the perturbed energy method.