Elliptic gradient estimates for a nonlinear heat equation and applications. (March 2017) - Wu, Jia-Yong
Abstract: In this paper, we study elliptic gradient estimates for a nonlinear f -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 f -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 f -heat equation. Some special cases are also discussed.