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arXiv · 1409.4821

Going beyond the threshold: scattering and blow-up in the focusing NLS equation

Thomas Duyckaerts·Svetlana Roudenko
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Abstract

We study the focusing nonlinear Schrödinger equation in the $L^2$-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 that threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrarily 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.

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Thomas Duyckaerts, Svetlana Roudenko. 2014-09-16. Going beyond the threshold: scattering and blow-up in the focusing NLS equation. https://doi.org/10.1007/s00220-014-2202-y

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