`a`
Communications on Pure and Applied Analysis (CPAA)
 

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

Pages: 63 - 81, Volume 7, Issue 1, January 2008      doi:10.3934/cpaa.2008.7.63

 
       Abstract        Full Text (257.0K)       Related Articles       

Yacheng Liu - College of Science, Harbin Engineering University, Harbin, 150001, China (email)
Runzhang Xu - College of Science, Harbin Engineering University, Harbin, 150001, China (email)

Abstract: 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, potential wells; global existence, nonexistence.
Mathematics Subject Classification:  Primary: 35M20, 35Q72.

Received: October 2006;      Revised: May 2007;      Available Online: October 2007.