  We consider a three-parameter family of non-linear equations with
$(p+1)-$order non-linearities. Such family includes as a particular member the
well-known $b-$equation, which encloses the famous Camassa-Holm equation. For
certain choices of the parameters we establish a global existence result and
show a scenario that prevent the wave breaking of solutions. Also, we explore
unique continuation properties for some values of the parameters.
