Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation

Abstract

<p style='text-indent:20px;'>In this article, we study the initial-value problem for inhomogeneous fractional nonlinear Schrödinger equation <p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1">\begin{document}$ i\partial_{t}u = (-\Delta)^{s}u-|x|^{-b}|u|^{2\sigma}u, \, \, \, (t, x)\in \mathbb{R} \times \mathbb{R}^{N}, $\end{document}</tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \frac{1}{2}&lt;s&lt;1, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \frac{2s-b}{N}\leq \sigma&lt;\frac{2s-b}{N-2s} $\end{document}</tex-math></inline-formula>. We prove a Gagliardo-Nirenberg-type estimate and use it to establish sufficient conditions for global existence in <inline-formula><tex-math id="M4">\begin{document}$ H^{s}(\mathbb{R}^{N}) $\end{document}</tex-math></inline-formula>. In addition, we derive a localized Virial estimate for inhomogeneous fractional nonlinear Schrödinger equation in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>, which uses Balakrishnan's formula for the fractional Laplacian <inline-formula><tex-math id="M6">\begin{document}$ (-\Delta)^{s} $\end{document}</tex-math></inline-formula> from semigroup theory. By these estimates, we give the blowup criterion of radial solutions in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-critical, <inline-formula><tex-math id="M9">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-supercritical and <inline-formula><tex-math id="M10">\begin{document}$ H^{s} $\end{document}</tex-math></inline-formula>-subcritical power.