Long time upper bounds for solutions of Leibenson's equation on Riemannian manifolds

We consider on Riemannian manifolds the Leibenson equation $${\mathrm{\partial }}_{t}u={\mathrm{\Delta }}_p{u}^q\mathrm{.}$$ We prove that a certain upper bound for weak solutions of this equation is equivalent to a euclidean-type Sobolev inequality.