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

TL;DR

This paper proves global well-posedness for the defocusing generalized Boussinesq equation in low-order Sobolev spaces using the I-method, extending previous results for the case when k=1.

Contribution

It extends the global well-posedness results to a broader class of equations with higher nonlinearities in low-regularity Sobolev spaces.

Findings

Global well-posedness established for s > 1 - 1/(3k)

Utilizes the I-method to handle low regularity

Extends previous results for k=1 to general k ≥ 1

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 $k=1$.