Paper · Mar 1, 2016

Elliptic gradient estimates for a nonlinear heat equation and applications

Jia-Yong Wu

math.DGarXiv:1603.00166
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Abstract

In this paper, we study elliptic gradient estimates for a nonlinear -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 -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 -heat equation. Some special cases are also discussed.

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Submitted
Mar 1, 2016
Subjects
math.DG
Journal
Nonlinear Anal. TMA, 151 (2017), 1-17

Comments: 20 pages

Ricci solitonsAncient solutionsPerelman entropyLimit spaces & RCD

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Gradient estimates and Liouville type theorems for Poisson equations (Nguyen Thac Dung and Nguyen Ngoc Khanh, 2018)6Dung 2018 New differential Harnack inequalities for nonlinear heat equations (Jia-Yong Wu, 2018)1Wu 2018 Sharp upper diameter bounds for compact shrinking Ricci solitons (Jia-Yong Wu, 2020)3Wu 2020 Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition (Ling Wu, 2022)4Wu 2022 Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials (Ali Taheri and Vahideh Vahidifar, 2023)2Taheri 2023 Willmore-type inequalities for closed hypersurfaces in weighted manifolds (Guoqiang Wu and Jia-Yong Wu, 2024)5Wu 2024 A rigidity result for ancient Ricci flows (Qi S. Zhang, 2024)7Zhang 2024 Heat kernel on Ricci shrinker metric measure spaces (Bing Wang and Jie Wang, 2026)8Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. New differential Harnack inequalities for nonlinear heat equations

    Jia-Yong Wu

    Same author Jia-Yong Wu · Shared terms “nonlinear heat”, “heat equation”

  2. Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

    Ali Taheri and Vahideh Vahidifar

    Topics Limit spaces & RCD · Shared terms “gradient estimates”, “smooth metric”, “metric measure”

  3. Sharp upper diameter bounds for compact shrinking Ricci solitons

    Jia-Yong Wu

    Topics Perelman entropy, Ricci solitons · Same author Jia-Yong Wu · Shared terms “Sobolev”

  4. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “heat equation”, “Émery Ricci”, “Bakry Émery”

  5. Willmore-type inequalities for closed hypersurfaces in weighted manifolds

    Guoqiang Wu and Jia-Yong Wu

    Topics Limit spaces & RCD, Ricci solitons · Same author Jia-Yong Wu · Shared terms “Émery Ricci”, “Bakry Émery”, “gradient Ricci”

  6. Gradient estimates and Liouville type theorems for Poisson equations

    Nguyen Thac Dung and Nguyen Ngoc Khanh

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “gradient estimates”, “smooth metric”, “Liouville”

  7. A rigidity result for ancient Ricci flows

    Qi S. Zhang

    Topics Perelman entropy, Ancient solutions · Shared terms “log Sobolev”

  8. Heat kernel on Ricci shrinker metric measure spaces

    Bing Wang and Jie Wang

    Topics Perelman entropy, Limit spaces & RCD, Ricci solitons · Shared terms “metric measure”, “measure spaces”, “Bakry Émery”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.