Long time behaviors for the inhomogeneous NLS with a potential in $\mathbb{R}^3$

TL;DR

This paper investigates the long-term behavior of solutions to a focusing inhomogeneous nonlinear Schrödinger equation with a potential in three-dimensional space, establishing scattering results that extend previous work to non-radial cases.

Contribution

The paper generalizes existing scattering results for the inhomogeneous NLS with potential to include non-radial solutions in three dimensions.

Findings

Established a scattering criterion for the equation.

Proved scattering results using Morawetz estimates.

Extended previous results to non-radial solutions.

Abstract

In this article, we aim to study the scattering of the solution to the focusing inhomogeneous nonlinear Schr\"odinger equation with a potential of form \begin{align*} i\partial_t u+\Delta u- Vu=-|x|^{-b}|u|^{p-1}u \end{align*} in the energy space $H^1(\R^3)$. We prove a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory, which generalizes the results of Dinh \cite{DD} to the non-radial symmetric case.