On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Abstract

We undertake a comprehensive study for the fractional nonlinear Schrodinger equation \begin{document}$ i\partial_t u - (-\Delta)^s u = \mu_1 |u|^{\alpha_1} u + \mu_2 |u|^{\alpha_2} u, \quad u(0) = u_0, $\end{document} where \begin{document}$ \frac{d}{2d-1} \leq s , \begin{document}$ 0 . Firstly, we establish the local and global well-posedness results for non-radial and radial \begin{document}$ H^s $\end{document} initial data, radial \begin{document}$ \dot{H}^{s_c}\cap \dot{H}^s $\end{document} initial data, where \begin{document}$ s_c = \frac{d}{2}-\frac{2s}{\alpha_2} $\end{document} . Secondly, we study the asymptotic behavior of global radial \begin{document}$ H^s $\end{document} solutions. Of particular interest is the \begin{document}$ L^2 $\end{document} -critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the \begin{document}$ L^2 $\end{document} -critical fractional nonlinear Schrodinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radial \begin{document}$ \dot{H}^{s_c} \cap \dot{H}^s $\end{document} solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with \begin{document}$ L^2 $\end{document} -critical and \begin{document}$ L^2 $\end{document} -subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both \begin{document}$ L^2 $\end{document} -critical and \begin{document}$ L^2 $\end{document} -supercritical cases, including mass-concentration and limiting profile.