Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 211, pp. 1-12.

Title: Cross-constrained problems for nonlinear Schrodinger
       equation with harmonic potential
        
Authors: Runzhang Xu (Harbin Engineering Univ., China)
         Chuang Xu (Univ. of Alberta, Alberta, Canada)

Abstract:
 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.