Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrödinger Equation in $H^s({\mathbb{R}}^n)$

Abstract

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation

where $0<s<\min \{n,\frac{n}{2}+1\}$, $0<b<\min \{2,n-s,1+\frac{n-2s}{2}\}$ and $f(u)$ is a nonlinear function that behaves like $\lambda |u{|}^{\sigma }u$ 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 Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249–286] by extending the validity of $s$ and $b$. In addition, we also have the small data scattering result.