On the Cauchy problem for the Schrödinger-Hartree equation

Abstract

In this paper, we undertake a comprehensive study for the Schrödinger-Hartree equation\begin{equation*}iu_t +\Delta u+ \lambda (I_\alpha \ast |u|^{p})|u|^{p-2}u=0,\end{equation*}where $I_\alpha$ is the Riesz potential. Firstly, we address questionsrelated to local and global well-posedness, finite time blow-up. Secondly, we derive the best constant of a Gagliardo-Nirenberg type inequality. Thirdly, the mass concentration is established for all the blow-up solutions in the $L^2$-critical case. Finally, the dynamics of the blow-up solutions with critical mass is in detail investigated in terms of the ground state.