On the existence of a global solution of a hyperbolic problem

A quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For a simplified formulation, a criterion for the existence of a global-in-time smooth solution is obtained. For the original system, a sufficient condition for singularity formation is found, and a sufficient condition for the smoothness of the solution within the nonrelativistic period of oscillations is established. In addition, it is shown that arbitrarily small perturbations of the trivial solution lead to the formation of singularities in a finite time. The results can be used to construct and substantiate numerical algorithms for modeling the breaking of plasma oscillations.