Asymptotic stability of a nonlinear wave for an outflow problem of the bipolar Navier–Stokes–Poisson system under large initial perturbation

In this paper, we study the time-asymptotic behavior of solutions to an outflow problem for the one-dimensional bipolar Navier–Stokes–Poisson system in the half space. First, we make some suitable assumptions on the boundary data and space-asymptotic states such that the time-asymptotic state of the solution is a nonlinear wave, which is the superposition of the transonic stationary solution and the 2-rarefaction wave. Next, we show the stability of this nonlinear wave under a class of large initial perturbation, provided that the strength of the transonic stationary solution is small enough, while the amplitude of the 2-rarefaction wave can be arbitrarily large. The proof is completed by a delicate energy method and a continuation argument. The key point is to derive the positive upper and lower bounds of the particle densities.