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 logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in context of manifolds, where metric potential evolve under Perelman-Ricci type flow. For use is made entropy monotonicity arguments ultracontractivity expressed terms optimal constant Sobolev inequality. Some interesting consequences these are presented discussed.