Elliptic gradient estimates for a nonlinear heat equation and applications
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文摘
In this paper, we study elliptic gradient estimates for a nonlinear ff-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton’s and Souplet–Zhang’s gradient estimates for positive solutions to the nonlinear ff-heat equation only assuming the ∞∞-Bakry–Émery Ricci tensor is bounded below. As applications, we prove parabolic Liouville properties for some kind of ancient solutions to the nonlinear ff-heat equation. Some special cases are also discussed.