In this paper, we mainly study the weakly dissipative peakon equations with higher-order nonlinearity. Under the effect of dissipation, we first derive the infinite propagation speed if the initial datum has a nonnegative compact support. Furthermore, we obtain the large-time behavior of the support of momentum density with the initial data compactly supported. The corresponding results are obtained by using some prior estimates and the energy method. It is worth noting that we need to overcome the difficulties caused by the high-order nonlinear structure and the dissipative effect of the equation. The obtained results generalize the previous results to a certain degree.

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In this paper, we study the weakly dissipative asymptotically equivalent shallow water wave equations on the circle, which is called the weakly dissipative periodic b-family of equations. Under the effect of dissipation, we derive two blow-up criteria to this equation with certain initial data. When the parameters α, λ, and b belong to a suitable range, our present blow-up criteria improve and extend the previous relevant blow-up results.

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In this paper, we mainly devote to investigate the periodic Dullin–Gottwald–Holm equation. By overcoming the difficulties caused by the complicated mixed nonlinear structure, a very useful priori estimate is derived in Lemma 2.7. Based on Hα1-conservation and L∞-estimate of solution, some new blow-up phenomena are derived for the periodic Dullin–Gottwald–Holm equation under different initial conditions.