Juha Kinnunen, Niko Marola, Michele Miranda, Jr., Fabio ParonettoSource: Adv. Differential Equations, Volume 17, Number 9-10, 801--832.Abstract:
In this paper we study problems related to parabolic partial differential
equations in metric measure spaces equipped with a doubling measure and
supporting a Poincaréinequality. 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.