The plasma density profile modification by radiation pressure near the critical surface has been extensively studied for normally incident light within the context of the hydrodynamic equations and the wave equation. In one-dimensional spherical geometry, the radial stability of these profiles to density perturbations is studied and it is shown that, except for a small range of parameters, the only stable profiles are accompanied by a transition from subsonic flow inside the critical-density surface to supersonic flow outside the critical-density surface. The long-time behaviour of the class of solutions that do not admit such transitions has been studied and that class exhibits repetitive, unsteady flow.