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 $u_{tt}-u_{xx}+u_{xxxx}-(|u{|}^{2k}u{)}_{xx}=0$, $k\ge 1$, on the real line is globally well-posed in $H^s(\mathbb{R})$ 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 $k=1$.

13 pages. References updated

References in corpus (1)

• Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation