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

We show that the Cauchy problem for the defocusing generalized Boussinesq equation $$ u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0, quad kgeq 1, $$ 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.