Exponential energy decay of solutions for an integro-differential equation with strong damping

The initial boundary value problem for an integro-differential equation with strong damping in Ω×(0,∞)Ω×(0,∞):utt−M(‖∇u‖22)Δu+∫0tg(t−s)Δu(s)ds−Δut=f(u), is considered, where Ω is a bounded domain with a smooth boundary ∂Ω. The asymptotic behavior of solutions is discussed under some conditions on g. Decay estimates of the energy function of solutions are also given, via an integral inequality introduced by Komornik (1994) [11].