Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces

Ming  Wang

doi:10.3934/dcds.2016053

Article Contents

2016, Volume 36, Issue 10: 5763-5788. Doi: 10.3934/dcds.2016053

This issue Previous Article The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension Next Article Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations

Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces

Ming Wang^1,

1.
School of Mathematics and Physics, China University of Geosciences, Wuhan 430074

Received: July 31, 2015

Revised: January 31, 2016

Published: May 2017

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Abstract

The global well-posedness of the BBM equation is established in $H^{s,p}(\textbf{R})$ with $s\geq \max\{0,\frac{1}{p}-\frac{1}{2}\}$ and $1\leq p<\infty$. Moreover, the well-posedness results are shown to be sharp in the sense that the solution map is no longer $C^2$ from $H^{s,p}(\textbf{R})$ to $C([0,T];H^{s,p}(\textbf{R}))$ for smaller $s$ or $p$. Finally, some growth bounds of global solutions in terms of time $T$ are proved.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A01.

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