Small data global well--posedness and scattering for the inhomogeneous nonlinear Schr\"{o}dinger equation in $H^{s} (\mathbb R^{n})$

TL;DR

This paper establishes global well-posedness and scattering for the inhomogeneous nonlinear Schrödinger equation in Sobolev spaces with small initial data, extending previous results by broadening the parameter ranges for regularity and inhomogeneity.

Contribution

It extends the known conditions for global well-posedness of the INLS equation in Sobolev spaces, covering a wider range of parameters than prior work.

Findings

Proves global well-posedness in H^s for small initial data.

Establishes small data scattering results.

Extends validity of parameters s and b beyond previous limits.

Abstract

We consider the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f\left(u\right), u\left(0\right)=u_{0} \in H^{s} (\mathbb R^{n}),\] where $0<s<\min \left\{n,\;\frac{n}{2} +1\right\}$, $0<b<\min \left\{2,\;n-s,{\rm \; 1}+\frac{n-2s}{2} \right\}$ and $f\left(u\right)$ is a nonlinear function that behaves like $\lambda \left|u\right|^{\sigma } u$ with $\lambda \in \mathbb C$ and $\sigma >0$. We prove that the Cauchy problem of the INLS equation is globally well--posed in $H^{s} (\mathbb R^{n})$ if the initial data is sufficiently small and $\sigma _{0} <\sigma <\sigma _{s} $, where $\sigma _{0} =\frac{4-2b}{n} $ and $\sigma _{s} =\frac{4-2b}{n-2s} $ if $s<\frac{n}{2} $; $\sigma _{s} =\infty $ if $s\ge \frac{n}{2} $. Our global well--posedness result improves the one of Guzm\'{a}n in (Nonlinear Anal. Real World Appl. 37: 249--286, 2017) by extending the validity of $s$ and $b$. In addition, we also have the small data scattering result.