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May  2009, 8(3): 913-922. doi: 10.3934/cpaa.2009.8.913

Sharp threshold of global existence for the generalized Davey-Stewartson system in $R^2$

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

1. 

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

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

Received  May 2008 Revised  October 2008 Published  February 2009

This paper is concerned with the generalized Davey-Stewartson system in $\mathbf R^2$ which appears as mathematical models for the evolution of shallow-water waves having one predominant direction of travel. We obtain a sharp threshold of blowing up and global existence to the Cauchy problem of the system by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Furthermore, we answer the question: How small are the initial data, the global solutions to the Cauchy problem of the system exist.
Keywords: sharp threshold, cross-invariant manifold, Davey-Stewartson system, blowup., global existence.
Mathematics Subject Classification: Primary: 35A15, 35Q55; Secondary: 35B3.
Citation: Zaihui Gan, Boling Guo, Jian Zhang. Sharp threshold of global existence for the generalized Davey-Stewartson system in $R^2$. Communications on Pure and Applied Analysis, 2009, 8 (3) : 913-922. doi: 10.3934/cpaa.2009.8.913
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