Well-posedness and peakons for a higher-order $\mu$-Camassa-Holm equation

In this paper, we study the Cauchy problem of a higher-order $\mu$-Camassa-Holm equation. By employing the Green's function of $(\mu-\partial_{x}^{2})^{-2}$, we obtain the explicit formula of the inverse function $(\mu-\partial_{x}^{2})^{-2}w$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we prove the existence of global strong solutions and weak solutions. Moreover, we show that the data-to-solution map is H\"{o}lder continuous in $H^{s}(\mathbb{S})$, $s\geq 4$, equipped with the $H^{r}(\mathbb{S})$-topology for $0\leq r<s$. Finally, the equation is shown to admit single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.