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Parabolic Gradient Estimates and Harnack Inequalities for a Nonlinear Equation Under The Ricci Flow

Parabolic Gradient Estimates and Harnack Inequalities for a Nonlinear Equation Under The Ricci Flow When the Riemannian metric evolves under the Ricci flow, we investigate parabolic gradient estimates (Li–Yau’s type and J. Li’s type) for positive solutions to the non- |∇u| linear parabolic equation ( − ∂ )u = (p + 1) + qu on the underlying manifold. Based on these gradient estimates, we derive associated Harnack inequalities, respec- tively. Keywords Parabolic gradient estimate · Nonlinear parabolic equation · Ricci flow Mathematics Subject Classification 35B45 · 35K55 · 53C44 1 Introduction In 1980s, Li and Yau (1986) derived a gradient estimate, which was known as the Li–Yau estimate, for the heat equation on a complete Riemannian manifold. More- over, they deduced Harnack inequalities. The Harnack inequality also applied to the Ricci flow by Hamilton (1993) and played an important role in solving the Poincaré conjecture Cao and Zhu (2006); Perelman (Perelman). After the fundamental work of Li and Yau (1986), the investigation of Li–Yau estimates for general parabolic partial differential equations of second order has drawn much attentions. To name a few, Yang (2008) proved the Li–Yau estimate for the equation on a Riemannian manifold of ∂ u = u + au ln u + bu with a, b ∈ R, which was http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

Parabolic Gradient Estimates and Harnack Inequalities for a Nonlinear Equation Under The Ricci Flow

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References (19)

Publisher
Springer Journals
Copyright
Copyright © 2020 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-019-00193-6
Publisher site
See Article on Publisher Site

Abstract

When the Riemannian metric evolves under the Ricci flow, we investigate parabolic gradient estimates (Li–Yau’s type and J. Li’s type) for positive solutions to the non- |∇u| linear parabolic equation ( − ∂ )u = (p + 1) + qu on the underlying manifold. Based on these gradient estimates, we derive associated Harnack inequalities, respec- tively. Keywords Parabolic gradient estimate · Nonlinear parabolic equation · Ricci flow Mathematics Subject Classification 35B45 · 35K55 · 53C44 1 Introduction In 1980s, Li and Yau (1986) derived a gradient estimate, which was known as the Li–Yau estimate, for the heat equation on a complete Riemannian manifold. More- over, they deduced Harnack inequalities. The Harnack inequality also applied to the Ricci flow by Hamilton (1993) and played an important role in solving the Poincaré conjecture Cao and Zhu (2006); Perelman (Perelman). After the fundamental work of Li and Yau (1986), the investigation of Li–Yau estimates for general parabolic partial differential equations of second order has drawn much attentions. To name a few, Yang (2008) proved the Li–Yau estimate for the equation on a Riemannian manifold of ∂ u = u + au ln u + bu with a, b ∈ R, which was

Journal

Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals

Published: Jan 3, 2020

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