On the Cauchy problem for a higher-order μ-Camassa-Holm equation

Feng Wang; Fengquan Li; Zhijun Qiao

doi:10.3934/dcds.2018181

Article Contents

2018, Volume 38, Issue 8: 4163-4187. Doi: 10.3934/dcds.2018181

This issue Previous Article Global regularity for the 2D micropolar equations with fractional dissipation Next Article Bifurcation of limit cycles for a family of perturbed Kukles differential systems

On the Cauchy problem for a higher-order μ-Camassa-Holm equation

1.
School of Mathematics and Statistics, Xidian University, Xi'an 710071, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
3.
School of Mathematical & Statistical Sciences, University of Texas-Rio Grande Valley, Texas 78539, USA

Authors to whom correspondence should be addressed. E-mails: wangfeng@xidian.edu.cn, fqli@dlut.edu.cn, zhijun.qiao@utrgv.edu.

Received: November 30, 2017

Revised: January 31, 2018

Published: May 2018

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Abstract

In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. We first establish the Green's function of $(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^{s}(\mathbb{S})$, $s≥ 4$. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

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Mathematics Subject Classification: 35G25, 35L05, 35B30.

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