Gradient estimates and Harnack inequalities for a Yamabe-type parabolic equation under the Yamabe flow
type: Item
instance of: scholarly article
title: Gradient estimates and Harnack inequalities for a Yamabe-type parabolic equation under the Yamabe flow
zbMATH Open document ID: 1469.35069
author: Liangdi Zhang
published in: Science China. Mathematics
publication date: 2021-07-08T00:00:00Z
review text: The gradient estimate is an essential tool in understanding solutions of nonlinear partial differential equations from geometry. In 1980s, Li and Yau proved the well-known Li-Yau type estimate to the heat equation \((\Delta-\partial_t)u = 0\). Based on this, they deduced Harnack inequalities. The Harnack inequality was also applied to the Ricci flow by Hamilton and played an important role in solving the Poincaré conjecture. Hamilton proved an elliptic type gradient estimate, which is known as the Hamilton type gradient estimate, and showed that the Harnack estimate of Li and Yau is the trace of a full matrix inequality. Kotschwar later generalized the Hamilton type gradient estimate to the noncompact case. In 2018, Dung gave a global Hamilton type gradient estimate for a Yamabe type parabolic equation of \((\Delta-\partial_t)u = + au + bu^\alpha\) on Riemannian manifolds with the Ricci curvature bounded from below. In this paper, the author extends the previous results deriving a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation \[(\Delta- \partial_t)u = pu + qu^{a+1}\] under the Yamabe flow. Here \(p, q \in C^{2,1} (M^n \times [0, T])\) and \(a\) is a positive constant.
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 53E99
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35R01
zbMATH DE Number: 7369611
zbMATH Keywords: gradient estimate
zbMATH Keywords: Harnack inequality
zbMATH Keywords: Yamabe-type parabolic equation
zbMATH Keywords: Yamabe flow
MaRDI profile type: Publication
full work available at URL: s11425-019-1596-3
OpenAlex ID: W3007941277
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DOI: 10.1007/S11425-019-1596-3
description: scientific article; zbMATH DE number 7369611