ИСТИНА

We consider a quasilinear system of hyperbolic equations that describes plane electrostatic non-relativistic oscillations of electrons in a cold plasma with allowance for electron-ion collisions and show that it is equivalent to the pressureless Euler-Poisson equations with nonzero background density in the presence of relaxation. In 1D case we obtain a criterion for the existence of a global in time smooth solution to the Cauchy problem. It allows to accurately separate the initial data into two classes: one corresponds to a globally in time smooth solutions, and the other leads to a finite-time blowup. The influence of electron collision frequency $ \nu $ on the solution is investigated. It is shown that there is a threshold value, after exceeding which the regime of damped oscillations is replaced by the regime of monotonic damping. The set of initial data corresponding to a globally in time smooth solution of the Cauchy problem expands with increasing $ \nu $, however, at an arbitrarily large value there are smooth initial data for which the solution forms a singularity in a finite time, and this time tends to zero as $ \nu $ tends to infinity. The character of the emerging singularities is illustrated by numerical examples.