Elliptic gradient estimates for a nonlinear heat equation and applications. (March 2017)

• Record Type: Journal Article
• Authors: Wu, Jia-Yong
• Is Part Of: Nonlinear analysis. Volume 151(2017)
• Page Start: 1
• Page End: 17
• Publication Date: 2017-03
• Subjects: primary 53C21 58J35 -- secondary 35B53 35K55; Gradient estimate -- Liouville theorem -- Smooth metric measure space -- Bakry–Émery Ricci tensor -- Log-Sobolev inequality; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.11.014
• Languages: English
• ISSNs: 0362-546X

Abstract

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.