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

Zaihui Gan; Boling Guo; Jian Zhang

doi:10.3934/cpaa.2009.8.913

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2009, Volume 8, Issue 3: 913-922. Doi: 10.3934/cpaa.2009.8.913

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

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: April 30, 2008

Revised: September 30, 2008

Published: April 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 35A15, 35Q55; Secondary: 35B30.

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

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