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We prove that arbitrary small smooth perturbations of the zero equilibriumstate of the repulsive pressureless Euler-Poisson equations, which describe thebehavior of cold plasma, blow up for any non-constant doping profile already inone-dimensional space. We also propose a numerical procedure that allows one tofind the blow-up time for any initial data and present examples of suchcalculations for various doping profiles for standard initial data,corresponding to the laser pulse.

The repulsive Euler-Poisson equations with variable doping profile