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Differential Harnack estimates for parabolic equations

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Complex and Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

Let (M,g(t)) be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type

$$\frac{\partial}{\partial t}f=\Delta f- f{\rm ln}f+Rf.$$

We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.

Mathematics Subject Classification (2010) 53C44.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    Xiaodong Cao

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Zhou Zhang

Authors
  1. Xiaodong Cao
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  2. Zhou Zhang
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Corresponding author

Correspondence to Xiaodong Cao .

Editor information

Editors and Affiliations

  1. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

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Cao, X., Zhang, Z. (2011). Differential Harnack estimates for parabolic equations. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_5

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