Local and global well-posedness in $L^{2}(\mathbb R^{n})$ for the inhomogeneous nonlinear Schr\"{o}dinger equation

This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $iu_{t} +\Delta u=\lambda \left|x\right|^{-b} \left|u\right|^{\sigma } u, u(0)=u_{0} \in L^{2}(\mathbb R^{n})$, where $\lambda \in \mathbb C$, $0