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

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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

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

Received October 2006 Revised May 2007 Published October 2007

Abstract
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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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