Asymptotic to systems with memory and non-local initial data

We study the existence and the asymptotic behavior of the solution of an abstract viscoelastic system submitted to non-local initial data. utt + Au -a0tg(t - s)Bu(s)ds = 0 u(0) = ζ(u)in V,ut(0) = I (u)in H, where A and B are differential operators satisfying B ≈ Aα for 0 ≤ α ≤ 1. We prove that the model is well-posed. Concerning the asymptotic behavior, we show that the exponential decay holds if and only if α = 1 and g goes to zero exponentially. Otherwise if 0 ≤ α < 1 or the kernel goes to zero polynomially, then the solution only decays polynomially. We show the optimality of our result. Finally, we consider the non-dissipative case.