  In this paper, persistence properties of solutions are investigated for a
4-parameter family ($k-abc$ equation) of evolution equations having
$(k+1)$-degree non-linearities 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 $c$.
