Harnack estimates for nonlinear heat equations with potentials in geometric flows

Let M be a closed Riemannian manifold with a family of Riemannian metrics $${g_{\it ij}(t)}$$gij(t) evolving by geometric flow (Formula Presnted) is a family of smooth symmetric two-tensors on M. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat equation with potential: (Formula Presnted) ,where $${\gamma (t)}$$γ(t) is a continuous function on t, a is a constant and (Formula Presnted) is the trace of (Formula Presnted). Our Harnack estimates include many known results as special cases, and moreover lead to new Harnack inequalities for a variety geometric flows.