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Li-Yau Harnack Estimates for a Heat-Type Equation Under the Geometric Flow

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Abstract

In this paper, we consider the gradient estimates for a postive solution of the nonlinear parabolic equation ∂tu = Δtu + hup on a Riemannian manifold whose metrics evolve under the geometric flow ∂tg(t) = − 2Sg(t). To obtain these estimate, we introduce a quantity \(\underline {\boldsymbol {S}}\) along the flow which measures whether the tensor Sij satisfies the second contracted Bianchi identity. Under conditions on Ricg(t),Sg(t), and \(\underline {\boldsymbol {S}}\), we obtain the gradient estimates.

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Acknowledgments

The authors thank for Professor Kefeng Liu’s constant guidance and help, and also referee’s useful comments.

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Authors and Affiliations

  1. Faculty of Science, Technology and Communication (FSTC), Mathematic Research Unit, Campus Belval, Universite du Luxembourg, Maison du Nombre, 6, Avenue de la Fonte, L-4364, Esch-sur-Alzette, Luxembourg

    Yi Li

  2. China Maritime Police Academy, Ningbo, 315801, China

    Xiaorui Zhu

Authors
  1. Yi Li
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  2. Xiaorui Zhu
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Corresponding author

Correspondence to Yi Li.

Additional information

Yi Li is partially supported by the Fonds National de la Recherche Luxembourg (FNR) unde the OPEN scheme (project GEOMREV O14/7628746). Xiaorui Zhu is partially supported by Natural Science Foundation of China (grant) No. 11601091.

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Li, Y., Zhu, X. Li-Yau Harnack Estimates for a Heat-Type Equation Under the Geometric Flow. Potential Anal 52, 469–496 (2020). https://doi.org/10.1007/s11118-018-9739-x

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  • DOI: https://doi.org/10.1007/s11118-018-9739-x

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