Elliptic gradient estimates for a nonlinear f-heat equation on weighted manifolds with evolving metrics and potentials

We develop local elliptic gradient estimates for a basic nonlinear f-heat equation with a logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in the context of weighted manifolds, where the metric and potential evolve under a Perelman-Ricci type flow. For the heat bounds use is made of entropy monotonicity arguments and ultracontractivity estimates with the bounds expressed in terms of the optimal constant in the logarithmic Sobolev inequality. Some interesting consequences of these estimates are presented and discussed.