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.

arXiv comment13 pages

References in corpus (4)

• Algebraic bright and vortex solitons in defocusing media
• Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation
• Scattering for the radial focusing INLS equation in higher dimensions
• Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation