Well-posedness and peakons for a higher-order μ-Camassa–Holm equation. (October 2018)

• Record Type: Journal Article
• Authors: Wang, Feng; Li, Fengquan; Qiao, Zhijun
• Is Part Of: Nonlinear analysis. Volume 175(2018)
• Page Start: 210
• Page End: 236
• Publication Date: 2018-10
• Subjects: Higher-order μ-Camassa–Holm equation -- Global existence -- Weak solutions -- Hölder continuity -- Peakon solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.06.001
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study the Cauchy problem of a higher-order μ -Camassa–Holm equation. By employing the Green's function of ( μ − ∂ x 2 ) − 2, we obtain the explicit formula of the inverse function ( μ − ∂ x 2 ) − 2 w and local well-posedness for the equation in Sobolev spaces H s ( S ), s > 7 2 . Then we prove the existence of global strong solutions and weak solutions. Moreover, we show that the data-to-solution map is Hölder continuous in H s ( S ), s ≥ 4, equipped with the H r ( S ) -topology for 0 ≤ r < s . Finally, the equation is shown to admit single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.