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

Abstract: 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 $b=\frac 53$ in the model, we obtain the ill-posedness in $B^1_{\infty,1}$ and $B^{\frac 32}_{2,q}$ with $q\in(1,+\infty]$ in the sense of norm inflation.