Persistence properties of a Camassa–Holm type equation with <b>(</b><i>n</i> + 1<b>)</b>-order non-linearities

Lower order conservation laws and symmetries of a family hyperbolic equations having the Camassa–Holm equation as particular member are obtained. We show that has two with zeroth characteristics its generated by translations in independent variables certain scaling, well some invariant solutions studied. Next, we consider persistence asymptotic properties for considered. In particular, analyze behavior large values spatial variable. if initial data exponential decaying, then such property persists any time long solution exists. Moreover, depending on variable further shares same behavior, it necessarily vanishes identically. Finally, prove unique continuation results equation.