Well-posedness and peakons for a higher-order μ-Camassa-Holm equation

In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. By employing the Green's function of (μ−∂x2)−2, we obtain the explicit formula of the inverse function (μ−∂x2)−2w and local well-posedness for the equation in Sobolev spaces Hs(S), s>72. 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 Hs(S), s≥4, equipped with the Hr(S)-topology for 0≤r