Decay properties of solutions to a 4-parameter family of wave equations

In this paper, persistence properties of solutions are investigated for a 4-parameter family (k−abc equation) of evolution equations having (k+1)-degree nonlinearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, Novikov and Fokas-Olver-Rosenau-Qiao equations. These properties will imply that strong solutions of the k−abc 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 k, a, b, and c.