ИСТИНА

We consider the classical Cauchy problem for a system of equations describing 3D arbitrary electrostatic oscillations of the cold plasma. We introduce an iteration procedure that allows estimating the blow-up time from below. This procedure is constructive provided one succeeds in obtaining a two-sided estimate of an additional quantity depending on the solution. For the particular case of two-dimensional initial data with radial symmetry, refined sufficient conditions for destruction and preservation of smoothness in the first period of oscillations are obtained. Moreover, we give the example of estimating the blow-up time for the data for which results of numerics exist and discuss the roughness of our estimate.