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Harnack estimates for nonlinear heat equations with potentials in geometric flows

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Abstract

Let M be a closed Riemannian manifold with a family of Riemannian metrics \({g_{\it ij}(t)}\) evolving by geometric flow \({\partial_{t}g_{\it ij} = -2{S}_{\it ij}}\), where \({S_{\it ij}(t)}\) is a family of smooth symmetric two-tensors on M. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat equation with potential:

$$\frac{\partial f}{\partial t} = {\Delta}f + \gamma (t)\,{f}\,{\rm log}\,f +aSf,$$

where \({\gamma (t)}\) is a continuous function on t, a is a constant and \({S=g^{\it ij}S_{\it ij}}\) is the trace of \({S_{\it ij}}\). Our Harnack estimates include many known results as special cases, and moreover lead to new Harnack inequalities for a variety geometric flows.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035, Zhe-jiang, China

    Hongxin Guo

  2. Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

    Masashi Ishida

Authors
  1. Hongxin Guo
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  2. Masashi Ishida
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Correspondence to Masashi Ishida.

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Guo, H., Ishida, M. Harnack estimates for nonlinear heat equations with potentials in geometric flows. manuscripta math. 148, 471–484 (2015). https://doi.org/10.1007/s00229-015-0757-3

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  • DOI: https://doi.org/10.1007/s00229-015-0757-3

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