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

Abstract

<p style='text-indent:20px;'>We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta u = |x|^{-b} f(u), \;u(0)\in H^{s} (\mathbb R^{n} ), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\in \mathbb N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;s&lt;\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;b&lt;\min \{ 2, \;n-s, \;1+\frac{n-2s}{2} \} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is a nonlinear function that behaves like <inline-formula><tex-math id="M5">\begin{document}$ \lambda |u|^{\sigma } u $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \sigma&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \lambda \in \mathbb C $\end{document}</tex-math></inline-formula>. Recently, the authors in [<xref ref-type="bibr" rid="b1">1</xref>] proved the local existence of solutions in <inline-formula><tex-math id="M8">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ 0\le s&lt;\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula>. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in <inline-formula><tex-math id="M10">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ 0&lt; s&lt;\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula> doesn't follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial data in the standard sense in <inline-formula><tex-math id="M12">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula>, i.e. in the sense that the local solution flow is continuous <inline-formula><tex-math id="M13">\begin{document}$ H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula>, if <inline-formula><tex-math id="M14">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> satisfies certain assumptions.</p>