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In this paper, we investigate the Cauchy problem for the  N  –  abc  family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov,  Blow - up phenomena for the weakly dissipative Dullin - Gottwald - Holm equation , J. Math. Phys.  54  (2013), no. 9, 092703, DOI 10.1063/1.4820786] and [Z.Y. Zhang, J.H. Huang, and M.B. Sun,  Blow - up phenomena for the weakly dissipative Dullin - Gottwald - Holm equation revisited , J. Math. Phys.  56  (2015), no. 9, 092701, DOI 10.1063/1.4930198].

A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion