Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 39, pp. 1-18.

Title: Blow-up criteria and instability of standing waves for the inhomogeneous
       fractional Schrodinger equation
       
Authors: Binhua Feng (Northwest Normal Univ., Lanzhou, China)
         Zhiqian He (Qinghai Univ., Xining, China)
         Jiayin Liu (North Minzu Univ., Yinchuan, China)

Abstract: 
 In this article, we study the blow-up and instability of standing waves for the
 inhomogeneous fractional Schrodinger equation
 $$
 i\partial_tu-(-\Delta)^su+ |x|^{-b}|u|^{p}u=0,
 $$
 where $s\in (\frac{1}{2},1)$, $0<b<\min \{2s,N\}$ and $0<p< \frac{4s-2b}{N-2s}$.
 In the L<sup>2</sup>-critical and L<sup>2</sup>-supercritical cases, i.e.,
 $\frac{4s-2b}{N}\leq p< \frac{4s-2b}{N-2s}$, we establish general blow-up criteria
 for non-radial solutions by using localized virial estimates. Based on these
 blow-up criteria, we prove the strong instability of standing waves.

Submitted November 28, 2020. Published May 07, 2021.
Math Subject Classifications: 35B35, 35B40, 35K57, 35Q92, 92C17.
Key Words: Inhomogeneous fractional Schrodinger equation; blow-up criteria;
           strong instability.