General decay of the solution energy in a viscoelastic equation with a nonlinear source

In a bounded domain, we consider utt−Δu+∫0tg(t−τ)Δudτ=|u|γu, where γ>0and gg is a nonnegative and decaying function. We prove that, for certain class of relaxation functions and certain initial data, the rate of decay of energy is similar to that of gg. This result improves earlier ones in the literature in which only the exponential and polynomial decay rates are considered.