On the Global Existence and Blowup Phenomena of Schrödinger Equations with Multiple Nonlinearities

Details On the Global Existence and Blowup Phenomena of Schrödinger Equations with Multiple Nonlinearities On the Global Existence and Blowup Phenomena of Schrödinger Equations with Multiple Nonlinearities

2010-09-02

arxiv.org/abs/1009.0328v1

Mathematics

Analysis of PDEs

24 pages

Scientific paper

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}.

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