No abstract

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation iu_t +\Delta u=\lvert x\rvert^{-b} f(u),\quad u(0)=u_0 \in H^s(\mathbb{R}^n), where 0<s<\min\{n,\frac{n}{2} +1\} , 0<b<\min\{2,n-s,1+\frac{n-2s}{2}\} and f(u) is a nonlinear function that behaves like \lambda \lvert u\rvert^{\sigma} u with \lambda \in \mathbb{C} and \sigma >0 . We prove that the Cauchy problem of the INLS equation is globally well-posed in H^s(\mathbb{R}^n) if the initial data is sufficiently small and \sigma_0 <\sigma <\sigma_s , where \sigma_0 =\frac{4-2b}{n} and \sigma_s =\frac{4-2b}{n-2s} if s<\frac{n}{2} , \sigma_s =\infty if s\ge \frac{n}{2} . Our global well-posedness result improves the one of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249–286] by extending the validity of s and b . In addition, we also have the small data scattering result.

<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>

<p style='text-indent:20px;'>In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta u-c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t, x)\in \mathbb R\times\mathbb R^{d}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ d\ge3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;b&lt;2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \frac{4-2b}{d}&lt;\sigma&lt;\frac{4-2b}{d-2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ c&gt;-c(d):=-\left(\frac{d-2}{2}\right)^{2} $\end{document}</tex-math></inline-formula>. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzmán (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).</p>

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equa-where n ∈ N, 0 < s < min{n, 1 + n/2}, 0 < b < min{2, n − s, 1 + n−2s 2 } and f (u) is a nonlinear function that behaves like λ |u| σ u with σ > 0 and λ ∈ C. Recently, An-Kim [1] proved the local existence of solutions in H s (R n ) with 0 ≤ 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 (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 (R n ), i.e. in the sense that the local solution flow is continuous H s (R n ) → H s (R n ), if σ satisfies certain assumptions.