Existence and stability of stationary solution to compressible Navier-Stokes-Poisson equations in half line

In this paper, we investigate the asymptotic stability of the stationary solution to the outflow problem for the compressible Navier-Stokes-Poisson system in a half line. We show the existence of the stationary solution with the aid of the stable manifold theory. The time asymptotic stability of the stationary solution is obtained by the elementary energy method. Furthermore, for the supersonic flow at spatial infinity, we also obtain an algebraic and an exponential decay rate, when the initial perturbation belongs to the corresponding weighted Sobolev space. The proof is based on a time and space weighted energy method.