Persistence properties and unique continuation for a generalized Camassa-Holm equation

In this paper, persistence properties of solutions are investigated for a generalized Camassa-Holm equation (g-kbCH) having (k+1)-degree nonlinearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, and the Novikov equations. The persistence properties will imply that strong solutions of the g-kbCH equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters b and k. Finally, existence of global solutions is established when b = k+1.