On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Van Duong Dinh; Binhua Feng

doi:10.3934/dcds.2019188

Article Contents

2019, Volume 39, Issue 8: 4565-4612. Doi: 10.3934/dcds.2019188

This issue Previous Article 2D incompressible Euler equations: New explicit solutions Next Article Chern-Simons gauged sigma model into

$ \mathbb{H}^2 $

and its self-dual equations

On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Van Duong Dinh^1,2, and
Binhua Feng^3, ,

1.
Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2.
Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam
3.
Department of Mathematics, Northwest Normal University, Lanzhou, China

^* Corresponding author: Binhua Feng
^* Corresponding author: Binhua Feng

Received: September 2018

Revised: February 2019

Published online: August 2019

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Abstract

We undertake a comprehensive study for the fractional nonlinear Schrödinger equation

$ i\partial_t u - (-\Delta)^s u = \mu_1 |u|^{\alpha_1} u + \mu_2 |u|^{\alpha_2} u, \quad u(0) = u_0, $

where $ \frac{d}{2d-1} \leq s <1 $, $ 0 < \alpha_1 <\alpha_2 < \frac{4s}{d-2s} $. Firstly, we establish the local and global well-posedness results for non-radial and radial $ H^s $ initial data, radial $ \dot{H}^{s_c}\cap \dot{H}^s $ initial data, where $ s_c = \frac{d}{2}-\frac{2s}{\alpha_2} $. Secondly, we study the asymptotic behavior of global radial $ H^s $ solutions. Of particular interest is the $ L^2 $-critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the $ L^2 $-critical fractional nonlinear Schrödinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radial $ \dot{H}^{s_c} \cap \dot{H}^s $ solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with $ L^2 $-critical and $ L^2 $-subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both $ L^2 $-critical and $ L^2 $-supercritical cases, including mass-concentration and limiting profile.

Keywords:

Fractional nonlinear Schrödinger equation,
global existence,
scattering,
blow-up criteria,
blow-up dynamics.

Mathematics Subject Classification: Primary: 35B44; Secondary: 35Q55.

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