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arXiv · 1603.00166

Elliptic gradient estimates for a nonlinear heat equation and applications

Jia-Yong Wu
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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-\'Emery 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.

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Jia-Yong Wu. 2016-03-01. Elliptic gradient estimates for a nonlinear heat equation and applications. https://arxiv.org/abs/1603.00166

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