Abstract
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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