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.

arXiv comment20 pages

References in corpus (3)

• Algebraic bright and vortex solitons in defocusing media
• Scattering of radial solutions to the Inhomogeneous Nonlinear Schrödinger Equation
• Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation