Asymptotic to systems with memory and non-local initial data

Abstract

We study the existence and the asymptotic behavior of the solution of an abstract viscoelastic system submitted to non-local initial data. u(tt )+ Au - integral(t)(0) g(t - s)Bu(s)ds = 0 u(0) = xi(u) in V, u(t) (0) = eta(u) in H, where A and B are differential operators satisfying B approximate to A(alpha) for 0 <= alpha <= 1. We prove that the model is well-posed. Concerning the asymptotic behavior, we show that the exponential decay holds if and only if alpha = 1 and g goes to zero exponentially. Otherwise if 0 <= a < 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.