Instantaneous blow-up versus local solvability for one problem of propagation of nonlinear waves in semiconductors

This work develops the theory of the blow-up phenomena for one Sobolev problem that arises in the theory of propagation of nonlinear waves in semiconductors. This problem is considered as 1) the Cauchy problem, 2) the initial-boundary value problem on the half-line and 3) the initial-boundary value problem on a segment. It was shown that in the first two cases the problem does not have weak solution even locally in time, but in the third case the problem has the classical solution that exists at least locally in time. The upper estimate of solvability time for classical solution in the third case is obtained. This analytical a priori information was used in the numerical experiment, which is able to determine the process of the solution's blow-up more accurately.