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

TL;DR

The paper analyzes the Cauchy problem for the fifth-order Camassa–Holm (FOCH) equation, deriving sharp local well-posedness thresholds in Besov spaces that depend on the nonlinearity parameter $b$, establishing a blow-up criterion and conserved-quantity-based global existence for $0\le b\le 1$, and proving ill-posedness at $b=\frac{5}{3}$ via norm inflation in $B^1_{\infty,1}$ and $B^{\frac{3}{2}}_{2,q}$. The methodology combines Littlewood–Paley theory, transport estimates, and Lagrangian-coordinate techniques to obtain precise regularity requirements, a robust global existence framework, and explicit ill-posedness mechanisms. These results highlight the delicate interplay between dispersion, nonlinearity, and the higher-order structure of FOCH, with implications for peakon dynamics and long-time behavior. The work advances the understanding of well-posedness and instability regimes for higher-order Camassa–Holm-type models in Besov scales.

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.

Theorems & Definitions (33)

• Proposition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Lemma 2.6
• Proposition 2.7
• Lemma 2.8
• Lemma 2.9
• Lemma 2.10