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

Abstract

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\] where $n\in \mathbb{N}$, $0<s<\min \{n,1+n/2\}$, $0<b<\min \{2,n-s,1+\frac{n-2s}{2}\}$ and $f(u)$ is a nonlinear function that behaves like $\lambda {\left|u\right|}^{\sigma }u$ with $\sigma >0$ and $\lambda \in \mathbb{C}$. Recently, An--Kim \cite{AK21} proved the local existence of solutions in $H^s({\mathbb{R}}^n)$ with $0\le s<\min \{n,1+n/2\}$. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in $H^s({\mathbb{R}}^n)$ with $0<s<\min \{n,1+n/2\}$ doesn't follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial data in the standard sense in $H^s({\mathbb{R}}^n)$, i.e. in the sense that the local solution flow is continuous $H^s({\mathbb{R}}^n)\to H^s({\mathbb{R}}^n)$, if $\sigma$ satisfies certain assumptions.

31 pages

References in corpus (1)

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