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

On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows