Asymptotic stability of stationary waves to the Navier-Stokes-Poisson equations in half line

The main concern of this paper is to investigate the asymptotic stability of stationary solution to the compressible Navier-Stokes-Poisson equations with the classical Boltzmann relation in a half line. We first show the unique existence of stationary solution with the aid of the stable manifold theory, and then prove that the stationary solution is time asymptotically stable under the small initial perturbation by the elementary energy method. Finally, we discuss the convergence rate of the time-dependent solution towards the stationary solution, and give a new condition to ensure an algebraic decay or an exponential decay. The proof is based on a time and space weighted energy method by fully utilizing the self-consistent Poisson equation.