Motivated by recent research achievement of quantum interacting systems in non-equilibrium, we consider a Luttinger model with a suddenly switched-on interaction proposed by Cazalilla [M.A. Cazalilla, Phys. Rev. Lett. 97 (2006) 156403]. In order to compare with real systems, we extend Cazalilla's scenario to the spinful system. To find the influence of initial states on the time evolution of some non-equilibrium systems, we mainly focus on the density-density propagator. By comparison and analysis, we discover the different behavior of this non-equilibrium system. Further, it is found that the propagator saves strong memory of initial state, and the effects of right-left interaction cancel out in total density-density propagator.