Well-posedness and peakons for a higher-order μ-Camassa–Holm equation. (October 2018) - Wang, Feng; Li, Fengquan; Qiao, Zhijun
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.