In this paper, we consider the Cauchy's problem of global existence and scattering behavior of small, smooth, and localized solutions of cubic fractional Schr\"odinger equations in one dimension, \begin{ equation*} \mathrm{ i} \partial_t u- (-\Delta)^{ \frac{ \alpha}{ 2}} u=c_*|u|^2u, \end{ equation*} where $\alpha \in (\frac{ 1}{ 3},1), c_* \in \mathbb{ R}$. Our work is a generalization of the result due to Ionescu and Pusateri \cite{ IP}, where the case $\alpha=\frac{ 1}{ 2}$ was considered. The highlight in this paper is to give a modified dispersive estimate in weighted Sobolev spaces for cubic fractional Schr\"odinger equations, which could be used for $ \alpha \in (\frac{ 1}{ 3},1)$. Based on this modified dispersive estimate, we prove the global existence and modified scattering behavior of solutions combining space-time resonance and bootstrap arguments.
Categories: math.AP