On the Global Existence and Blowup Phenomena of Schr\"{o}dinger Equations with Multiple Nonlinearities

In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=\Delta u-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0, &u(x,0)=u_0(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $V(x)$ and $W(x)$ are real-valued potentials with $V(x)\geq 0$ and $W$ is even, $f(x,|u|^2)$ is measurable in $x$ and continuous in $|u|^2$, and $u_0(x)$ is a complex-valued function of $x$. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem. These results can be looked as the supplement to Chapter 6 of \cite{Cazenave2}. In addition, our results extend those of \cite{Zhang} and improve some of \cite{Tao2}.