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

We study the focusing nonlinear Schrödinger equation i∂tu+Δu+|u|p-1u=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i\partial_t u +\Delta u + |u|^{p-1}u=0}$$\end{document}, x∈RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x \in \mathbb{R}^N}$$\end{document} 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.