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A new rotation-two-component -Hunter–Saxton system is considered in the spatially periodic setting. The local well-posedness for the system is established by Kato theory. Two wave-breaking criteria and the sufficient conditions of blow-up solution are derived by relying on the Riccati differential equation and localization analysis. In addition, the Lyapunov function method is used to prove the existence of global solutions under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived. [ABSTRACT FROM AUTHOR]

The existence of global weak solutions for a dissipative model equation for shallow water wave of moderate amplitude is studied in the spacewithout the sign condition on the initial value by employing the limit technique of viscous approximation. A new one-sided lower bound and the higher integrability estimate act a key role in our analysis. Our results partly extend the work of Coclite et al. on the existence of global weak solutions to the generalized hyperlastic-rod equation. [ABSTRACT FROM PUBLISHER]