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This article studies a nonlinear parabolic equation on a complete weighted manifold where the metric and potential evolve under a super Perelman-Ricci flow. It derives elliptic gradient estimates of local and global types for the positive solutions and exploits some of their implications notably to a general Liouville type theorem, parabolic Harnack inequalities and classes of Hamilton type dimension-free gradient estimates. Some examples and special cases are discussed for illustration.

Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications