Global solutions for the generalized Boussinesq equation in low-order Sobolev spaces

We show that the Cauchy problem for the defocusing generalized Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(\mathrm{\mid }u{\mathrm{\mid }}^{2k}u{)}_{xx}=0$, $k\ge 1$, on the real line is globally well-posed in $H^s(\mathbb{R})$ for $s>1-(1\mathrm{/}3k)$. We use the “$I$-method” to define a modification of the energy functional that is “almost conserved” in time. Our result extends the previous one obtained by Farah and Linares (2010 $\mathit{\text{J.}}\mathit{\text{London}}\mathit{\text{Math.}}\mathit{\text{Soc.}}$ $\mathbf{\text{81}}$ 241-254) when $k=1$.