On the Cauchy problem for a class of cubic quasilinear shallow-water equations.

In this paper, we mainly investigate the Cauchy problem of the cubic Camassa-Holm-type equations which arise as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves, which admit the single peaked solitons and multi-peakon solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model. Then, by overcoming the difficulties cased by the complicated mixed nonlinear structure, we present a precise blow-up scenario. Moreover, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions for Cauchy problem in the Gevrey-Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. Finally, we investigate the persistence properties of the solution to the Cauchy problem, we prove that the solution maintains the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]