Global existence and blow-up for the focusing inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential

TL;DR

This paper investigates the conditions under which solutions to a focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential exist globally or blow up, extending previous results to more general initial data and thresholds.

Contribution

It establishes new criteria for global existence and blow-up for the equation, including non-radial and infinite variance solutions, broadening the understanding of solution behavior.

Findings

Derived criteria for global existence and blow-up.

Extended previous results to general initial data.

Analyzed solutions relative to the ground state threshold.

Abstract

In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential \[iu_{t} +\Delta u-c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0<b<2$, $\frac{4-2b}{d}<\sigma<\frac{4-2b}{d-2}$ and $c>-c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzm\'{a}n (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).