On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Binhua Feng

doi:10.3934/cpaa.2018085

Article Contents

2018, Volume 17, Issue 5: 1785-1804. Doi: 10.3934/cpaa.2018085

This issue Previous Article Positive radial solutions of a nonlinear boundary value problem Next Article Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Binhua Feng^,

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

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

Received: May 2017

Revised: December 2017

Published: April 2018

This work is supported by NSFC Grants (No. 11601435, No. 11475073), Gansu Provincial Natural Science Foundation (1606RJZA010) and NWNU-LKQN-14-6.

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Abstract

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

$i\partial_t u-(-Δ)^su+λ_1|u|^{2p_1}u+λ_2|u|^{2p_2}u = 0, $

where $ 0<p_1<p_2<\frac{2s}{N-2s}$ . Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case $ 0<p_1<\frac{2s}{N}$ and $p_2 = \frac{2s}{N}$ . Finally, we investigate the dynamical properties of blow-up solutions, including $L^2$-concentration, blow-up rate and limiting profile.

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Mathematics Subject Classification: 35Q51, 35Q55.

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