In this talk, the global escape phenomenon of gas-like transonic fluids with self-gravitation in spherically symmetric space-times is studied. The escape phenomenon is governed by an initial-boundary problem of one-dimensional compressible Euler-Poisson equations which form a mixed-type nonlinear partial differential system of balance laws. The compressible Euler-Poisson system is reformulated as a 3x3 hyperbolic system of balance laws by the equations of fluid's density and the gravitational potential. The global existence to the shock wave solutions of fluid's density-momentum and the Lipschitz continuous solution to the gradient of potential, is established by a new version of the generalized Glimm scheme (GGS for short). In this talk, we also investigate the steady-state solution of Euler-Poisson equation in far field. The current study shows the decay of the density and potential are of certain order and the estimates are crucial to build up the approximate solutions in GGS.