The Cauchy problem for the critical inhomogeneous nonlinear Schr\"{o}dinger equation in $H^{s}(\mathbb R^{n})$

TL;DR

This paper investigates the well-posedness and scattering of solutions to the critical inhomogeneous nonlinear Schrödinger equation in fractional Sobolev spaces, extending understanding of solution behavior under specific inhomogeneity and nonlinearity conditions.

Contribution

The paper proves local and small data global well-posedness and scattering results for the critical INLS in $H^{s}$ spaces, employing fractional Hardy inequalities and Strichartz estimates.

Findings

Established local well-posedness for the critical INLS.

Proved small data global well-posedness and scattering.

Developed nonlinear estimates using fractional Hardy inequality.

Abstract

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f(u), ~u(0)=u_{0} \in H^{s} (\mathbb R^{n} ),\] where $n\ge3$, $1\le s<\frac{n}{2} $, $0<b<2$ and $f(u)$ is a nonlinear function that behaves like $\lambda \left|u\right|^{\sigma } u$ with $\lambda \in \mathbb C$ and $\sigma =\frac{4-2b}{n-2s} $. We establish the local well-posedness as well as the small data global well-posedness and scattering in $H^{s} (\mathbb R^{n} )$ with $1\le s<\frac{n}{2}$ for the critical INLS equation under some assumption on $b$. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.