Sharp conditions of global existence for nonlinear Schrodinger and Klein-Gordon equations

Global existence for the Cauchy problems of nonlinear Schrodinger and Klein-Gordon equations was presented. The relations between the global existence of the Cauchy problem of these equations and the ground state were observed. It was proved that the solutions of the Cauchy problem for the Schrodinger equation blow up in a finite time for a class of sufficiently large data and exist in all time for small initial data.