Remarks on the fractional inhomogeneous Hartree equation

TL;DR

This paper investigates the inhomogeneous fractional Schrödinger equation, establishing sharp thresholds for global existence and blow-up of solutions in certain regimes using adapted Gagliardo-Nirenberg inequalities.

Contribution

It introduces a novel threshold criterion for solution behavior in the inhomogeneous fractional Schrödinger equation using a specialized Gagliardo-Nirenberg inequality.

Findings

Sharp threshold for global existence versus blow-up.

Use of adapted Gagliardo-Nirenberg inequality.

Results applicable in mass super-critical and energy sub-critical regimes.

Abstract

This paper studies the inhomogeneous fractional Sch\"odinger equation $$i\dot u-(-\Delta)^s u=\pm(I_\alpha *|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to the above problem, the standing waves give a sharp threshold of global existence versus finite time blow-up of solutions.