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

Abstract

We show that the Cauchy problem for the defocusing generalized Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0$, $k\geq1$, on the real line is globally well-posed in $H^{s}(\R)$ for $s>1-({1}/{3k})$. 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 \textit{J. London Math. Soc.} \textbf{81} 241-254) when $k=1$.