We investigate the solutions behavior of some dissipative systems. Precisely, we study the following problems: 1. The Navier-Stokes equations with a time periodic external force in Rn. We show that a time periodic solution exists when the space dimension n ≥ 5 under some suitably small assumption. The main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator. With the optimal decay estimates, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem. In addition, we also study the time asymptotic stability of the time periodic solution. 2. Compressible Euler-Poisson system with nonlinear damping added to the momentum equations. Under some mild conditions, the solutions of the Cauchy problem for the system globally exist and converge to the nonlinear diffusion waves, which are the corresponding solutions of nonlinear parabolic equations given by Darcy's law with specified initial data. The optimal convergence rates are obtained by Green function method when the initial perturbation is in L1-space. 3. 2 x 2 linear damped p-system with boundary effect. By a heuristic analysis, we realize that the best asymptotic profile for the original solution is the parabolic solution of the IBVP for the corresponding porous media equation with a specified initial data. In particular, we further show the convergence rates of the original solution to its best asymptotic profile, which are much better than the rates obtained in previous works. The approach adopted in this part is the elementary weighted energy method together with Green function one.