Stability of transonic shocks to the Euler-Poisson system with varying background charges

This paper is devoted to studying the stability of transonic shock solutions to the Euler-Poisson system in a one-dimensional nozzle of finite length. The background charge in the Poisson equation is a piecewise constant function. The structural stability of the steady transonic shock solution is obtained by the monotonicity argument. Furthermore, this transonic shock is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data. One of the crucial ingredients of the analysis is to establish the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions.