Long time behaviors for the inhomogeneous NLS with a potential in $\mathbb{R}^3$
Fanfei Meng, Sheng Wang, Chengbin Xu
TL;DR
This work analyzes the focusing inhomogeneous NLS with a potential in $\mathbb{R}^3$, proving global scattering for $0<b<1$ and $1+\frac{4-2b}{3}<p<1+4-2b$ under $V\in\mathcal{K}_0\cap L^{\frac{3}{2}}$ with $\|V_-\|_{\mathcal{K}_0}<4\pi$, and $x\cdot\nabla V\in L^r$ for $r\ge\tfrac{3}{2}$. The authors develop a non-radial scattering criterion and a Morawetz-type estimate accommodating a potential, then combine a variational analysis around the ground state $Q$ with a virial-Morawetz framework. A key novelty is replacing radial symmetry tools with specially designed cutoffs and exploiting the decay of $|x|^{-b}$ to handle non-radial data, thereby extending prior radial results (e.g., Dinh) to the non-radial setting and broadening the admissible gradient integrability of the potential. The results contribute to the understanding of long-time dynamics for intercritical INLS with external potentials and provide techniques potentially applicable to related dispersive systems.
Abstract
In this article, we aim to study the scattering of the solution to the focusing inhomogeneous nonlinear Schrödinger equation with a potential of form \begin{align*} i\partial_t u+Δu- 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.