Harnack estimates for nonlinear backward heat equations in geometric flows

Hongxin Guo; Masashi Ishida

doi:10.1016/j.jfa.2014.08.006

Harnack estimates for nonlinear backward heat equations in geometric flows

Hongxin Guo, Masashi Ishida

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

13 Citations (Scopus)

Abstract

Let M be a closed Riemannian manifold with a family of Riemannian metrics g_ij(t) evolving by a geometric flow ∂_tg_ij=-2S_ij, where S_ij(t) is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation f∂t=-δf+γflog f+aSf where a and γ are constants and S=g^ijS_ij is the trace of S_ij. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover leads to new Harnack inequalities for a variety of geometric flows.

Original language	English
Pages (from-to)	2638-2662
Number of pages	25
Journal	Journal of Functional Analysis
Volume	267
Issue number	8
DOIs	https://doi.org/10.1016/j.jfa.2014.08.006
Publication status	Published - 2014 Oct 15

Keywords

Geometric flows
Harnack estimates
Nonlinear heat equations
Ricci flow

ASJC Scopus subject areas

Analysis

Access to Document

10.1016/j.jfa.2014.08.006

Cite this

@article{37b1c685d1364204bdba07604c8ff700,

title = "Harnack estimates for nonlinear backward heat equations in geometric flows",

abstract = "Let M be a closed Riemannian manifold with a family of Riemannian metrics gij(t) evolving by a geometric flow ∂tgij=-2Sij, where Sij(t) is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation f∂t=-δf+γflog f+aSf where a and γ are constants and S=gijSij is the trace of Sij. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover leads to new Harnack inequalities for a variety of geometric flows.",

keywords = "Geometric flows, Harnack estimates, Nonlinear heat equations, Ricci flow",

author = "Hongxin Guo and Masashi Ishida",

note = "Funding Information: The first author was supported by NSFC (Grants No. 11001203 and 11171143 ) and Zhejiang Provincial Natural Science Foundation of China (Project No. LY13A010009 ). The second author was partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science , No. 25400074 . Publisher Copyright: {\textcopyright} 2014 Elsevier Inc.",

year = "2014",

month = oct,

day = "15",

doi = "10.1016/j.jfa.2014.08.006",

language = "English",

volume = "267",

pages = "2638--2662",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "8",

}

TY - JOUR

T1 - Harnack estimates for nonlinear backward heat equations in geometric flows

AU - Guo, Hongxin

AU - Ishida, Masashi

N1 - Funding Information: The first author was supported by NSFC (Grants No. 11001203 and 11171143 ) and Zhejiang Provincial Natural Science Foundation of China (Project No. LY13A010009 ). The second author was partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science , No. 25400074 . Publisher Copyright: © 2014 Elsevier Inc.

PY - 2014/10/15

Y1 - 2014/10/15

N2 - Let M be a closed Riemannian manifold with a family of Riemannian metrics gij(t) evolving by a geometric flow ∂tgij=-2Sij, where Sij(t) is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation f∂t=-δf+γflog f+aSf where a and γ are constants and S=gijSij is the trace of Sij. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover leads to new Harnack inequalities for a variety of geometric flows.

AB - Let M be a closed Riemannian manifold with a family of Riemannian metrics gij(t) evolving by a geometric flow ∂tgij=-2Sij, where Sij(t) is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation f∂t=-δf+γflog f+aSf where a and γ are constants and S=gijSij is the trace of Sij. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover leads to new Harnack inequalities for a variety of geometric flows.

KW - Geometric flows

KW - Harnack estimates

KW - Nonlinear heat equations

KW - Ricci flow

UR - http://www.scopus.com/inward/record.url?scp=85027948883&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027948883&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2014.08.006

DO - 10.1016/j.jfa.2014.08.006

M3 - Article

AN - SCOPUS:85027948883

SN - 0022-1236

VL - 267

SP - 2638

EP - 2662

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 8

ER -

Harnack estimates for nonlinear backward heat equations in geometric flows

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this