The Cauchy problem for a generalized fifth-order Camassa-Holm type equation with pseudo-peakons

This paper focuses on the Cauchy problem for the fifth-order Camassa-Holm type equation with $ k $-th nonlinearity: $ m_t + m_x u^{k+1} + b m u^k u_x = 0, m = (1 - \alpha^2 \partial_x^2)(1 - \beta^2 \partial_x^2) u $, periodic pseudo-peakons traveling wave solutions are found for this FOCHT equation with all values of $ k\in\mathbb{N} $ and $ b\in \mathbb{R},\alpha>0,\beta>0 $. It is known that the critical Sobolev index for the third-order Camassa-Holm type equation is $ 3/2 $. This leads us to conjecture that the optimal Sobolev index for the FOCHT equation should be $ 5/2 $. In this paper, another aim is to prove this conjecture through the following two aspects. We first show the local-in-time well-posedness for the solution of the FOCHT equation for the Cauchy data of the velocity $ u \in B_{p,1}^{2+\frac{1}{p}} $ with $ 1 \leq p < \infty $, then we demonstrate that this FOCHT equation is ill-posed in $ B^{\frac{5}{2}}_{2,\infty} $ by utilizing the pseudo-peakons solutions.