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

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