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A unified approach for energy scattering for focusing nonlinear Schrödinger equations

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  • We consider the Cauchy problem for focusing nonlinear Schrödinger equation

    $ i\partial_t u + \Delta u = - |u|^\alpha u, \quad (t, x) \in \mathbb R \times \mathbb R^N, $

    where $ N\geq 1 $, $ \alpha>\frac{4}{N} $ and $ \alpha <\frac{4}{N-2} $ if $ N\geq 3 $. We give a criterion for energy scattering for the equation that covers well-known scattering results below, at and above the mass and energy ground state threshold. The proof is based on a recent argument of Dodson-Murphy [Math. Res. Lett. 25(6):1805-1825, 2018] using the interaction Morawetz estimate.

    Mathematics Subject Classification: Primary: 35A01; Secondary: 35Q55.


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