On the Cauchy problem for a class of shallow water wave equations with (k + 1)-order nonlinearities

This paper considers the Cauchy problem for a class of shallow water wave equations with (k+1)(k+1)-order nonlinearities in the Besov spaces∂tu−∂t∂x2u=uk∂x3u+buk−1∂xu∂x2u−(b+1)uk∂xu, which involves the Camassa–Holm, the Degasperis–Procesi and the Novikov equations as special cases. Firstly, by means of the transport equation and the Littlewood–Paley theory, we obtain the local well-posedness of the equations in the nonhomogeneous Besov space Bp,rs (s>max⁡{1+1p,32} and p,r∈[1,+∞]p,r∈[1,+∞]). Secondly, we consider the local well-posedness in B2,rs with the critical index s=32, and show that the solutions continuously depend on the initial data. Thirdly, the blow-up criteria and the conservative property for the strong solutions are derived. Finally, with the help of a new Ovsyannikov theorem, we investigate the Gevrey regularity and analyticity of the solutions. Moreover, we get a lower bound of the lifespan and the continuity of the data-to-solution mapping.