Li-Yau Harnack Estimates for a Heat-Type Equation Under the Geometric Flow

In this paper, we consider the gradient estimates for a postive solution of the nonlinear parabolic equation partial differential (t)u = Delta(t)u + hu(p) on a Riemannian manifold whose metrics evolve under the geometric flow partial differential (t)g(t) = - 2S(g(t)). To obtain these estimate, we introduce a quantity (S) under bar along the flow which measures whether the tensor S-ij satisfies the second contracted Bianchi identity. Under conditions on Ric(g(t)),S-g(t), and (S) under bar, we obtain the gradient estimates.