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

Jinmyong An; Roesong Jang; Jinmyong Kim

doi:10.3934/dcdsb.2022111

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2023, Volume 28, Issue 2: 1046-1067. Doi: 10.3934/dcdsb.2022111

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

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

^*Corresponding author: Jinmyong Kim (jm.kim0211@ryongnamsan.edu.kp)
^*Corresponding author: Jinmyong Kim (jm.kim0211@ryongnamsan.edu.kp)

Received: August 31, 2021

Revised: March 31, 2022

Early access: June 2022

Published: February 2023

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Abstract

In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger 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-Guzmán (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).

Keywords:

Inhomogeneous nonlinear Schrödinger equation,
inverse-square potential,
global existence,
blow-up,
ground state.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A01, 35B44.

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