Nonlinear Evolutionary Schrödinger Equation in the Supercritical Case

We prove that for some initial data, solutions of the Cauchy problem for the nonlinear Schrödinger equation in the supercritical case are destroyed after a finite time, the exact value of which can be estimated from above. Lower bounds are obtained for the rate of destruction of the solution in some norms. A set of initial data is identified for which the solution of the Cauchy problem for the nonlinear Schrödinger equation in the supercritical case exists globally.