  In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type
dimension free Harnack inequality for the heat equation $\partial_t u=Lu$
associated with the time dependent Witten Laplacian on complete Riemannian
manifolds equipped with a variant of the $(K, m)$-super Perelman Ricci flows
and the $K$-super Perelman Ricci flows.
