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<s<\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0<b<\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>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<\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< s<\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>

A note on the $H^{s}$-critical inhomogeneous nonlinear Schrödinger equation

“…ε = 0. The standard continuous dependence in H s with s > 0 for the subcritical INLS equation (1.1) was studied in [25]. More precisely they obtained the following result.…”

“…Meanwhile, the local and global well-posedness for the INLS equation (1.1) in the fractional Sobolev space H s (R n ) have also been attracted a lot of interest in recent years. See [1,[24][25][26][27][28] and the references therein. The H s -subcritical case was studied by [1,24,25,26,28].…”

“…See [1,[24][25][26][27][28] and the references therein. The H s -subcritical case was studied by [1,24,25,26,28]. Guzmán [24] established the local and global well-posedness in H s (R n ) with 0 ≤ s ≤ min 1, n 2 for the subcritical INLS equation (1.1) with f (u) = λ|u| σ u.…”

Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

“…The INLS equation (1.2) arises in nonlinear optics for modeling the propagation of laser beam and it has been widely studied by many authors. See, for example, [1,2,3,4,5,9,17,20,21,30] and the references therein. The local and global well-posedness as well as blow-up and scattering in the energy space H 1 have been widely studied by many authors.…”

Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential

“…Moreover, when c = 0 and b = 0, we have the inhomogeneous nonlinear Schrödinger equation, denoted by INLS equation, which has also attracted a lot of interest in recent years (see e.g. [1,2,3,6,10] and the references therein).…”