Global solutions to the Cauchy problem of viscous shallow water equations in some classes of large data

In this paper, we consider the Cauchy problem of shallow water equations under partial smallness assumption of the initial data and obtain the global existence and decaying estimates. We apply Green's function method, together with the regularity criterion and the abstract bootstrap argument which have been widely used in the previous work. Our paper shows that based on the precise information about the different frequency parts in Green's function and the resulted time decaying structures, one could give a concise and clear proof of the global existence of the solution and also obtain its decaying estimates.