  We consider persistence properties of solutions for a generalised wave
equation including vibration in elastic rods and shallow water models, such as
the BBM, the Dai's, the Camassa-Holm, and the Dullin-Gottwald-Holm equations,
as well as some recent shallow water equations with the Coriolis effect. We
establish unique continuation results and exhibit asymptotic profiles for the
solutions of the general class considered. From these results we prove the
non-existence of non-trivial spatially compactly supported solutions for the
equation. As an aftermath, we study the equations earlier mentioned in light of
our results for the general class.
