Sharp Criteria of Scattering for the Fractional NLS

TL;DR

This paper establishes the precise threshold conditions for scattering and blow-up of solutions to the fractional nonlinear Schrödinger equation in the supercritical regime, extending understanding of solution behavior based on initial data and conserved quantities.

Contribution

It provides the sharp criteria distinguishing global scattering solutions from finite-time blow-up for the fractional NLS in the supercritical case, refining previous results with exact threshold conditions.

Findings

Identifies sharp threshold conditions for scattering.

Demonstrates conditions leading to finite-time blow-up.

Extends known results to fractional NLS in supercritical regime.

Abstract

In this paper, the sharp threshold of scattering for the fractional nonlinear Schr\"{o}dinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\frac{4s}{N}<p<1+\frac{4s}{N-2s}$, and $$ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\frac{s-s_c}{s_c}}E[Q], \ M[u_{0}]^{\frac{s-s_c}{s_c}}\| u_{0}\|^2_{\dot H^s}<M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$$ then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that if $1+\frac{4s}{N}<p<1+\frac{4s}{N-2s}$ and $$ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\frac{s-s_c}{s_c}}E[Q], \ M[u_{0}]^{\frac{s-s_c}{s_c}}\| u_{0}\|^2_{\dot H^s}>M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s},$$ then the corresponding solution $u(t)$ blows up in finite time, according to Boulenger, Himmelsbach, and Lenzmann's results in [2].