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Lower order conservation laws and symmetries of a family of hyperbolicequations having the Camassa-Holm equation as a particular member are obtained.We show that the equation has two conservation laws with zeroth ordercharacteristics and that its symmetries are generated by translations in theindependent variables and a certain scaling, as well as some invariantsolutions are studied. Next, we consider persistence and asymptotic propertiesfor the solutions of the equation considered. In particular, we analyse thebehaviour of the solutions of the equation for large values of the spatialvariable. We show that if the initial data has a certain asymptotic exponentialdecaying, then such property persists for any time as long as the solutionexists. Moreover, depending on the behaviour of the initial data for largevalues of the spatial variable and if for some further time the solution sharesthe same behaviour, then it necessarily vanishes identically. Finally, we proveunique continuation results for the solutions of the equation.

Persistence properties of a Camassa-Holm type equation with $(n+1)-$order non-linearities