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

Abstract

This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schrödinger (INLS) equation $i{u}_t+\Delta u=\lambda {\left|x\right|}^{-b}{\left|u\right|}^{\sigma }u,u(0)=u_0\in L^2({\mathbb{R}}^n)$, where $\lambda \in \mathbb{C}$, $0<b<\min \left\{2,n\right\}$ and $0<\sigma \le \frac{4-2b}{n}$. We prove the local well-posedness and small data global well-posedness of the INLS equation in the mass-critical case $\sigma =\frac{4-2b}{n}$, which have remained open until now. We also obtain some local well-posedness results in the mass-subcritical case $\sigma <\frac{4-2b}{n}$. In order to obtain the results above, we establish the Strichartz estimates in Lorentz spaces and use the contraction mapping principle based on Strichartz estimates.

13 pages

References in corpus (2)

• Scattering for the radial focusing INLS equation in higher dimensions
• Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation