On threshold phenomena for the Euler-Poisson equations

It is proved that the radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background, corresponding to cold plasma oscillations, blow up in many spatial dimensions for almost all initial data. Moreover, if a solution is globally smooth in time, then it is either affine or tends to affine as $t\to\infty$. This behavior is strikingly different from the behavior of solutions of the Euler-Poisson equations with zero background, as well as solutions of "attractive" Euler-Poisson equations with a non-zero background, where the initial data is divided into two sets of non-zero measure, one of which corresponds to globally smooth solutions.