Some Gradient Estimates for the Heat Equation on Domains and for an Equation by Perelman

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound. The idea is to combine the Nash and Davies heat kernel estimate with a Sobolev imbedding by Hebey, together with a Hamilton type gradient estimate.