Differential Harnack Estimates for Parabolic Equations

Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type $$\text{ppt}f=\mathrm{\Delta }f-f\ln f+Rf\mathrm{.}$$ We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.