A unified approach for energy scattering for focusing nonlinear Schrödinger equations

Van Duong Dinh

doi:10.3934/dcds.2020286

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2020, Volume 40, Issue 11: 6461-6491. Doi: 10.3934/dcds.2020286

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A unified approach for energy scattering for focusing nonlinear Schrödinger equations

Van Duong Dinh^1,2,

1.
Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2.
Department of Mathematics, HCMC University of Education, 280 An Duong Vuong, Ho Chi Minh, Vietnam

Received: March 2020

Published: July 2020

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Abstract

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.

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

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References

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