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

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.