On well posedness for the inhomogeneous nonlinear Schr\"odinger equation

TL;DR

This paper investigates the well-posedness of the inhomogeneous nonlinear Schrödinger equation, establishing local and global results for initial data in Sobolev spaces using Strichartz estimates and contraction mapping.

Contribution

It provides new well-posedness results for the INLS with initial data in Sobolev spaces, extending previous understanding of the equation's behavior.

Findings

Local well-posedness in $H^s$ for $0 \\leq s \\leq 1$

Global well-posedness under certain conditions

Use of Strichartz estimates for analysis

Abstract

The purpose of this paper is to study well-posedness of the initial value problem (IVP) for the inhomogeneous nonlinear Schr\"odinger 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 $0\leq s\leq 1$. To this end, we use the contraction mapping principle based on the Strichartz estimates related to the linear problem.