Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

Abstract

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger equation (IBNLS) \[iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $\lambda \in \mathbb R$, $d\in \mathbb N$, $0<s<\min \{2+\frac{d}{2},\frac{3}{2}d\}$ and $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $H^{s}(\mathbb R^{d})$ if $\frac{8-2b}{d}<\sigma< \sigma_{c}(s)$ and the initial data is sufficiently small, where $\sigma_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $\sigma_{c}(s)=\infty$ if $s\ge \frac{d}{2}$.