Abstract and Applied Analysis

Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

Abstract

We consider the following Cauchy problem: $-i{u}_{t}=\mathrm{\Delta }u-V\left(x\right)u+f\left(x,{|u|}^{\mathrm{2}}\right)u+\left(W\left(x\right)\mathrm{\star }{|u|}^{\mathrm{2}}\right)u,\mathrm{}\mathrm{}\mathrm{}\mathrm{}x\in {ℝ}^{N},\mathrm{}\mathrm{}\mathrm{}\mathrm{}t>\mathrm{0},u\left(x,\mathrm{0}\right)={u}_{\mathrm{0}}\left(x\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}x\in {ℝ}^{N},$ where $V\left(x\right)$ and $W\left(x\right)$ are real-valued potentials and $V\left(x\right)\ge \mathrm{0}$ and $W\left(x\right)$ is even, $f\left(x,|u{|}^{\mathrm{2}}\right)$ is measurable in $x$ and continuous in $|u{|}^{\mathrm{2}}$, and ${u}_{\mathrm{0}}\left(x\right)$ 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.

Article information

Source
Abstr. Appl. Anal. Volume 2013 (2013), Article ID 238410, 14 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512175

Digital Object Identifier
doi:10.1155/2013/238410

Mathematical Reviews number (MathSciNet)
MR3134172

Zentralblatt MATH identifier
1294.35128

Citation

An, Xiaowei; Li, Desheng; Song, Xianfa. Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation. Abstr. Appl. Anal. 2013 (2013), Article ID 238410, 14 pages. doi:10.1155/2013/238410. https://projecteuclid.org/euclid.aaa/1393512175

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