On well posedness for the inhomogeneous nonlinear Schrödinger equation

Abstract

The purpose of this paper is to study well-posedness of the initial value problem (IVP) for the inhomogeneous nonlinear Schrödinger equation (INLS) $$i{u}_t+\Delta u+\lambda |x{|}^{-b}|u{|}^{\alpha }u=0,$$ where $\lambda =\pm 1$ and $\alpha$, $b>0$. We obtain local and global results for initial data in $H^s({\mathbb{R}}^N)$, with $0\le s\le 1$. To this end, we use the contraction mapping principle based on the Strichartz estimates related to the linear problem.

arXiv comment36 pages