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

Binhua Feng

doi:10.3934/cpaa.2018085

Abstract

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] 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.

Highlights

In recent years, there has been a great deal of interest in using fractional Laplacians to model physical phenomena
Levy-like quantum mechanical paths, Laskin in [22, 23] used the theory of functionals over functional measure generated by the Levy stochastic process to deduce the following nonlinear fractional Schrodinger equation i∂tu = (−∆)su + f (u), (1.1)
In [9], we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for 0 < p2

Summary

Introduction

There has been a great deal of interest in using fractional Laplacians to model physical phenomena. In [32], Tao et al undertook a comprehensive study for the following nonlinear Schrodinger equation with combined power-type nonlinearities i∂tu + ∆u + λ1|u|2p1 u + λ2|u|2p2 u = 0, They addressed questions related to local and global well-posedness, finite time blow-up, and asymptotic behaviour. We are interested in sufficient conditions about the existence of blow-up solutions, sharp thresholds of blow-up and global existence, the dynamical properties of blow-up solutions, including L2-concentration, blow-up rates, and limiting profile To solve these problems, we mainly use the ideas from Boulenger et al [1] and Keraani [18]. The dynamical properties of blow-up solutions for the L2-critical nonlinear Schrodinger equation (1.4) have been discussed in [18] In these papers, the study of blow-up solutions relies heavily on the scaling invariance of (1.1) and (1.4).

Preliminaries

The existence of blow-up solutions

Dynamic of blow-up solutions in the case of L2-critical

L2 and