Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential

TL;DR

This paper establishes scattering and blow-up criteria for the 3D cubic focusing inhomogeneous nonlinear Schrödinger equation with a potential, extending previous results to a broader range of the parameter b and considering radial initial data.

Contribution

It extends scattering and blow-up criteria for the inhomogeneous NLS with potential to the case 0<b<1, including new results for radial initial data.

Findings

Global well-posedness and scattering for radial data under certain energy and mass conditions.

Blow-up results for initial data exceeding specific energy and mass thresholds.

Extension of previous results to the case 0<b<1].

Abstract

In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schr\"{o}dinger equation with a potential $$ iu_{t}+\Delta u-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{3}}, $$ where $0<b<1$. We first establish global well-posedness and scattering for the radial initial data $u_{0}$ in $H^{1}({\bf R}^{3})$ satisfying $M(u_{0})^{1-s_{c}}E(u_{0})^{s_{c}}<\mathcal{E}$ and $\|u_{0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}<\mathcal{K}$ provided that $V$ is repulsive, where $\mathcal{E}$ and $\mathcal{K}$ are the mass-energy and mass-kinetic of the ground states, respectively. Our result extends the results of Hong \cite{H} and Farah-Guzm$\acute{\rm a}$n \cite{FG1} with $b\in(0,\frac12)$ to the case $0<b<1$. We then obtain a blow-up result for initial data $u_{0}$ in $H^{1}({\bf R}^{3})$ satisfying $M(u_{0})^{1-s_{c}}E(u_{0})^{s_{c}}<\mathcal{E}$ and $\|u_{0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}>\mathcal{K}$ if $V$ satisfies some additional assumptions.0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}>\mathcal{K}$ if $V$ satisfies some additional assumptions.