In this note we consider problems related to parabolic partial differential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincaré inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the $p$-Laplacian. Moreover, we prove the sufficiency of the Grigoryan--Saloff-Coste theorem for general $p>1$ in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.
"On the Harnack inequality for parabolic minimizers in metric measure spaces." Tohoku Math. J. (2) 65 (4) 569 - 589, 2013. https://doi.org/10.2748/tmj/1386354296