Electron. J. Diff. Equ., Vol. 2012 (2012), No. 41, pp. 1-13.

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

Luiz G. Farah, Hongwei Wang

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.

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Luiz G. Farah
ICEx, Universidade Federal de Minas Gerais
Av. Ant&ocric;nio Carlos, 6627, Caixa Postal 702, 30123-970
Belo Horizonte-MG, Brazil
email: lgfarah@gmail.com
Hongwei Wang
Faculty of Science, Xi'an Jiaotong University
Xi'an 710049, China
Department of Mathematics, Xinxiang College
Xinxiang 453003, China
email: wang.hw@stu.xjtu.edu.cn

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