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

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Abstract

We study the focusing nonlinear Schrödinger equation \({i\partial_t u +\Delta u + |u|^{p-1}u=0}\), \({x \in \mathbb{R}^N}\) 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 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.

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Authors and Affiliations

  1. Université Paris 13, Paris, France

    Thomas Duyckaerts

  2. The George Washington University, Washington, USA

    Svetlana Roudenko

Authors
  1. Thomas Duyckaerts
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  2. Svetlana Roudenko
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Correspondence to Svetlana Roudenko.

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Communicated by W. Schlag

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Duyckaerts, T., Roudenko, S. Going Beyond the Threshold: Scattering and Blow-up in the Focusing NLS Equation. Commun. Math. Phys. 334, 1573–1615 (2015). https://doi.org/10.1007/s00220-014-2202-y

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  • DOI: https://doi.org/10.1007/s00220-014-2202-y

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