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

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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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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Keywords

  • Nonlinear parabolic equation
  • Harnack estimate
  • Geometric flow

Mathematics Subject Classification (2010)

  • Primary 53C44