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July  2009, 8(4): 1303-1312. doi: 10.3934/cpaa.2009.8.1303

Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials

Zaihui Gan 1, , Boling Guo 2, and  Jian Zhang 1,

1. 

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068
2. 

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088

Received  December 2007 Revised  August 2008 Published  March 2009

This paper is concerned with the Cauchy problem for the nonlinear Schrödinger equations with multiple potentials. Under some assumptions on these potentials, we first obtain some sufficient conditions of blowup according to the initial energy and the Cauchy initial data directly. We next establish a sufficient condition of global existence by using potential well method and the relation between the Cauchy data and the ground state. We finally answer the question that how small the Cauchy initial data need to be for global existence by using scaling argument.
Keywords: global existence., Nonlinear Schrödinger equations, multiple potentials, blowup.
Mathematics Subject Classification: Primary: 35A15, 35Q55; Secondary: 35B3.
Citation: Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303
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