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

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

2010-09-29

arxiv.org/abs/1009.5726v2

Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 41, pp. 1-13

Mathematics

Analysis of PDEs

13 pages. References updated

Scientific paper

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 $k=1$.

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