In this paper, we study the Cauchy problem for the weakly dissipative Dullin–Gottwald–Holm equation. We establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time.

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In this paper, we consider the dissipative Camassa-Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated.

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