Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential

Abstract

In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u+|x|^{-b} |u|^{σ} u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge 3$, $0<b<2$, $\frac{4-2b}{d}<\sigma <\frac{4-2b}{d-2}$ 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-Guzmán (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).

22 pages. arXiv admin note: text overlap with arXiv:2107.09826