  This paper proves the strong parabolic Harnack inequality for local weak
solutions to the heat equation associated with time-dependent (nonsymmetric)
bilinear forms. The underlying metric measure Dirichlet space is assumed to
satisfy the volume doubling condition, the strong Poincar\'e inequality, and a
cutoff Sobolev inequality. The metric is not required to be geodesic. Further
results include a weighted Poincar\'e inequality, as well as upper and lower
bounds for non-symmetric heat kernels.
