2013 Volume 65 Issue 4 Pages 569-589
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 Grigor'yan–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.
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