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

TL;DR

This paper proves that solutions to the inhomogeneous nonlinear Schrödinger equation depend continuously on initial data in the Sobolev space $H^{s}$, extending previous local existence results by establishing standard continuous dependence.

Contribution

It demonstrates the continuous dependence of solutions on initial data in $H^{s}$ for the INLS equation, under certain conditions on the nonlinearity exponent $\sigma$, which was not previously established.

Findings

Established continuous dependence in $H^{s}$ for the INLS equation

Extended local existence results to include standard continuous dependence

Identified conditions on $\sigma$ for continuous dependence

Abstract

We consider the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\] 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.