Large-Time Behavior of Momentum Density Support of a Family of Weakly Dissipative Peakon Equations with Higher-Order Nonlinearity

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.

A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term

“…Using the concavity approach, the blow-up of solutions was investigated. We also refer to [9][10][11][12][13][14][15], where the large-time behavior of solutions to nonlinear wave equations has been studied by the energy and concavity methods. The applications of fractional calculus are broad.…”