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.