Existence of solutions of degenerate parabolic equations with inhomogeneous density and growing data on manifolds

We consider doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume. The equation is inhomogeneous: indeed it contains a capacitary coefficient depending on the space variable, which we assume to decay at infinity. We prove existence of solutions for initial data growing at infinity in a suitable admissible class and some related estimates. We also prove, independently, a sup bound valid in the same geometrical setting for solutions which are a priori known to have compact support; the majorization depends on the size of the support.