One considers the damped semilinear viscoelastic wave equation uₜₜ-Δ u+α u+f(u)+∫₀ᵗg(t-τ )Δ u(τ )\, dτ +h(uₜ)=0\,\,\,in\,\,\,Ω× (0,∞ ), where Ω is any bounded or finite measure domain of R^ n, α≥ 0 and $f,h$ are power like functions. The existence of global regular and weak solutions is proved by means of the Faedo-Galerkin method and uniform decay rates of the energy are obtained following the perturbed energy method by assuming that g decays exponentially.
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Cavalcanti et al. (2002) studied this question.