Electron. J. Diff. Equ., Vol. 2012 (2012), No. 211, pp. 1-12.

Cross-constrained problems for nonlinear Schrodinger equation with harmonic potential

Runzhang Xu, Chuang Xu

This article studies a nonlinear Schodinger equation with harmonic potential by constructing different cross-constrained problems. By comparing the different cross-constrained problems, we derive different sharp criterion and different invariant manifolds that separate the global solutions and blowup solutions. Moreover, we conclude that some manifolds are empty due to the essence of the cross-constrained problems. Besides, we compare the three cross-constrained problems and the three depths of the potential wells. In this way, we explain the gaps in [J. Shu and J. Zhang, Nonlinear Shrodinger equation with harmonic potential, Journal of Mathematical Physics, 47, 063503 (2006)], which was pointed out in [R. Xu and Y. Liu, Remarks on nonlinear Schrodinger equation with harmonic potential, Journal of Mathematical Physics, 49, 043512 (2008)].

Submitted August 3, 2012. Published November 27, 2012.
Math Subject Classifications: 78A60, 35Q55.
Key Words: Cross-constrained problem; blow up; global existence; invariant manifold; harmonic potential.

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Runzhang Xu
Department of Applied Mathematics
Harbin Engineering University, 150001, China
email: xurunzh@yahoo.com.cn
Chuang Xu
Department of Mathematiccal and Statistical Sciences
University of Alberta, Edmonton T6G 2G1, Alberta, Canada
Department of Mathematics
Harbin Institute of Technology, 150001, China
email: xuchuang6305@163.com

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