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

$ 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 |u|^{\sigma } u $ with $ \sigma>0 $ and $ \lambda \in \mathbb C $. Recently, the authors in [1] 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.