  In this paper, we study the Cauchy problem for a generalized integrable
Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming
the difficulties caused by the complicated mixed nonlinear structure, we
firstly establish the local well-posedness result in Besov spaces, and then
present a precise blow-up scenario for strong solutions. Furthermore, we show
the existence of single peakon by the method of analysis.

Analysis of PDEs