Going Beyond the Threshold: Scattering and Blow-up in the Focusing NLS Equation

We study the focusing nonlinear Schrödinger equation $${i\partial_t u +\Delta u + |u|^{p-1}u=0}$$i∂tu+Δu+|u|p-1u=0, $${x \in \mathbb{R}^N}$$x∈RN in the L2-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass–energy) threshold. In our first result, we extend the known scattering versus blow-up dichotomy above the mass–energy threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrary large mass and energy. As a consequence, we characterize the behavior of the ground state initial data modulated by a quadratic phase. Our second result gives two blow up criteria, which are also applicable in the energy-supercritical NLS setting. We finish with various examples illustrating our results.