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

Abstract

Abstract 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].