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January  2008, 7(1): 63-81. doi: 10.3934/cpaa.2008.7.63

Potential well method for initial boundary value problem of the generalized double dispersion equations

Yacheng Liu 1, and  Runzhang Xu 1,

1. 

College of Science, Harbin Engineering University, Harbin, 150001, China, China

Received  October 2006 Revised  May 2007 Published  October 2007

In this paper we study the initial boundary value problem of the generalized double dispersion equations $u_{t t}-u_{x x}-u_{x x t t}+u_{x x x x}=f(u)_{x x}$, where $f(u)$ include convex function as a special case. By introducing a family of potential wells we first prove the invariance of some sets and vacuum isolating of solutions, then we obtain a threshold result of global existence and nonexistence of solutions. Finally we discuss the global existence of solutions for problem with critical initial condition.
Keywords: Generalized double dispersion equation, initial boundary value, nonexistence., potential wells; global existence.
Mathematics Subject Classification: Primary: 35M20, 35Q7.
Citation: Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63
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