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 \[iu_{t} +Δu=|x|^{-b} f\left(u\right), u\left(0\right)=u_{0} \in H^{s} (\mathbb R^{n}),\] where $0<s<\min \left\{n,\frac{n}{2}+1\right\}$, $0<b<\min \left\{2,n-s,1+\frac{n-2s}{2}\right\}$ 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 Guzmán 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.

21 pages

References in corpus (1)

• Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation