  We prove that arbitrary smooth perturbations of the zero equilibrium state of
the repulsive pressureless Euler-Poisson equations, which describe the behavior
of cold plasma, blow up for any non-constant doping profile already in
one-dimensional space. Further, we study small perturbations of the equilibrium
to determine which properties of the doping profile contribute to the blow-up.
We also propose a numerical procedure that allows one to find the blow-up time
for any initial data and present examples of such calculations for various
doping profiles for standard initial data, corresponding to the laser pulse.
