Global existence and blowup for Choquard equations with an inverse-square potential

TL;DR

This paper studies the global behavior of solutions to the Choquard equation with inverse-square potential, establishing well-posedness, and conditions for global existence or blowup in both focusing and defocusing regimes.

Contribution

It provides the first comprehensive analysis of Choquard equations with inverse-square potential, including well-posedness and global existence/blowup criteria.

Findings

Global well-posedness in $H^1( ^N)$ for all cases

Global existence in defocusing case for all initial data

Blowup vs. global existence dichotomy below ground state in focusing case

Abstract

In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in $H^1(\mathbb{R}^N)$ is established. Moreover, the global existence result for arbitrary initial values is proved in the defocusing case while a global existence/blowup dichotomy below the ground state is established in the focusing case.