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

Abstract

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u), ~u(0)=u_{0} \in H^{s} (\mathbb R^{n} ),\] where $n\ge 3$, $1\le s<\frac{n}{2}$, $0<b<2$ 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.

20 pages

References in corpus (1)

• Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation