In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale- and location-invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers.
"Harnack's inequality for parabolic De Giorgi classes in metric spaces." Adv. Differential Equations 17 (9/10) 801 - 832, September/October 2012.