Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 41, pp. 1-13.

Title: Global solutions in lower order Sobolev spaces  for the generalized 
       Boussinesq equation

Authors: Luiz G. Farah (Univ. Federal de Minas Gerais, Brazil)
         Hongwei Wang (Xi'an Jiaotong Univ., China)

Abstract:
 We show that the Cauchy problem for the defocusing generalized Boussinesq
 equation
 $$
 u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0, \quad k\geq 1,
 $$
 on the real line is globally well-posed in $H^s(\mathbb{R})$ with 
 s>1-(1/(3k)). To do this, we use the I-method, introduced by 
 Colliander, Keel, Staffilani,  Takaoka and Tao [8,9], 
 to define a modification of the energy  functional that is almost 
 conserved in time.  Our result extends a previous result obtained 
 by Farah and Linares [16] for the case k=1.
 
Submitted October 3, 2011. Published March 16, 2012.
Math Subject Classifications: 35B30, 35Q55.
Key Words: Boussinesq equation; global well-posedness; I-method.