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arXiv 1009.0328math.AP
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具多重非线性项的薛定谔方程解的整体存在性和爆破现象

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

Xianfa Song

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24 pages
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摘要：

本文考虑了如下Cauchy问题\begin{align*} \left\{\begin{array}{ll}&-i u_t=Δu-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*}的整体存在性和爆破现象，其中$V(x)$和$W(x)$是具有$V(x)\geq 0$的实值位势，$W$是偶数，$f(x,|u|^2)$在$x$中可测，在$|u|^2$中连续，$u_0(x)$是$x$的复值函数.我们得到了问题解的爆破和整体存在的充分条件和两个尖锐的阈值。这些结果可作为对\cite{Cazenave2}第6章的补充。此外，我们的结果推广了\cite{Zhang}的结果，改进了\cite{Tao2}的某些结果.

英文摘要：

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=Δu-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}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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