# Sharp Threshold of Blow-up and Scattering for the fractional Hartree equation

## TL;DR

This paper establishes a precise threshold distinguishing between scattering and blow-up for solutions of the fractional Hartree equation with radial data, based on conserved quantities and ground state properties.

## Contribution

It provides a sharp criterion for global existence and blow-up in the fractional Hartree equation, extending understanding of critical thresholds in nonlinear dispersive equations.

## Key findings

- Identifies sharp threshold conditions for scattering versus blow-up.
- Demonstrates the ground state solution as a critical threshold.
- Shows the conditions are optimal with explicit solitary wave solutions.

## Abstract

We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}<M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, then the solution $u(t)$ is globally well-posed and scatters; if $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}>M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, $Q$ is the ground-state solution for the fractional Hartree equation.