Blow-up or Grow-up for the threshold solutions to the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation with $L^2$-supercritical and $H^1$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.