  Based on the idea of a recent paper by Ambrosio-Gigli-Savar\'e in Invent.
Math. (2013), we show that flow of the $q$-Cheeger energy, called $q$-heat
flow, solves the gradient flow problem of the Renyi entropy functional in the
$p$-Wasserstein. For that, a further study of the $q$-heat flow is presented
including a condition for its mass preservation. Under a convexity assumption
on the upper gradient, which holds for all $q\ge2$, one gets uniqueness of the
gradient flow and the two flows can be identified. Smooth solution of the
$q$-heat flow are solution the parabolic q-Laplace equation, i.e.
$\partial_{t}f_{t}=\Delta_{q}f_{t}.$
