  We consider the Cauchy problem for doubly non-linear degenerate parabolic
equations on Riemannian manifolds of infinite volume, or in $\R^N$. The
equation contains a weight function as a capacitary coefficient which we assume
to decay at infinity. We connect the behavior of non-negative solutions to the
interplay between such coefficient and the geometry of the manifold, obtaining,
in a suitable subcritical range, estimates of the vanishing rate for long times
and of the finite speed of propagation. In supercritical ranges we obtain
universal bounds and prove blow up in a finite time of the (initially bounded)
support of solutions.
