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In this paper, we investigate the Cauchy problem for the weakly dissipative Dullin-Gottwald-Holm equation. We first establish the local well-posedness result by Kato’s semigroup theory. Then, we obtain the precise blow-up scenario and present some blow-up results for strong solutions to the equation. Finally, we discuss the blow-up rate of the wave-breaking solutions. This result complements the early one in the literature, such as Novruzov [J. Math. Phys. 54, 092703 (2013)].

For a Carnot group G of step two, we prove that H-convex functions are locally bounded from above. Therefore, H-convex functions on a Carnot group G of step two are locally Lipschitz continuous by using recent results by Magnani.

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