Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations
2D incompressible Euler equations: New explicit solutions
August 2019, 39(8): 4565-4612. doi: 10.3934/dcds.2019188
On fractional nonlinear Schrödinger equation with combined power-type nonlinearities
Van Duong Dinh 1,2, and Binhua Feng 3,,
Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam
Department of Mathematics, Northwest Normal University, Lanzhou, China
* Corresponding author: Binhua Feng
Received September 2018 Revised February 2019 Published May 2019
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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, $
$ \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
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
-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
-critical and
-subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both
-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.
Citation: Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
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