The repulsive Euler-Poisson equations with variable doping profile

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.