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In this paper we mainly investigate the Cauchy problem of a generalized Camassa–Holm equation. We first derive two global existence results and two blow‐up results from the relationship between the Degasperis–Procesi equation and the generalized Camassa–Holm equation. We then prove the existence and uniqueness of global weak solutions under some certain sign conditions on the initial data.

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In this paper, we first establish the local well-posedness and continuous dependence for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect in nonhomogeneous Besov spaces B s p,r with s > 1 + 1/p or s = 1 + 1/p, p ∈ [1, +∞), r = 1 by a new way: the compactness argument and Lagrangian coordinate transformation, which removes the index constraint s > 3/2 and improves our previous work [21]. Then, we prove the solution is not uniformly continuous dependence on the initial data in both supercritical and critical Besov spaces.

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