On the blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities

TL;DR

This paper investigates the conditions under which solutions to a fractional nonlinear Schrödinger equation blow up, identifying sharp thresholds and analyzing their dynamical properties such as concentration and blow-up rate.

Contribution

It provides new sufficient conditions for blow-up, sharp thresholds for global existence, and detailed analysis of blow-up dynamics in fractional nonlinear Schrödinger equations.

Findings

Established sufficient conditions for blow-up solutions.

Derived sharp thresholds for blow-up and global existence.

Analyzed blow-up dynamics including concentration and rate.

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.