The local well-posedness, global existence and ill-posedness for the fifth order Camassa-Holm model

In this paper, we consider the fifth order Camassa-Holm model. Firstly, we improve the local well-posedness results in \cite{TangLiu2015,FOCH2021}. Secondly, we give the blow up criteria and conditions for global existence. Finally, when $b=\frac{5}{3}$ in the model, we obtain the ill-posedness in $B_{\mathrm{\infty },1}^1$ and $B_{2,q}^{\frac{3}{2}}$ with $q\in (1,+\mathrm{\infty }]$ in the sense of norm inflation.