  In this paper, we consider the Cauchy problem for the generalized
Camassa-Holm equation proposed by Hakkaev and Kirchev (2005) \cite{Hakkaev
2005}. We prove that the solution map of the generalized Camassa-Holm equation
is not uniformly continuous on the initial data in Besov spaces. Our result
include the present work (2020) \cite{Li 2020,Li 2020-1} on Camassa-Holm
equation with $Q=1$ and extends the previous non-uniform continuity in Sobolev
spaces (2015) \cite{Mi 2015} to Besov spaces. In addition, the non-uniform
continuity in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R})$ is the first
to be considered in our paper.
