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Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces

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Abstract

Given a complete, smooth metric measure space \((M,g,e^{-f}dv)\) with the Bakry–Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general f-heat equations

$$\begin{aligned} u_t=\Delta _f u+au\log u+bu +Au^p+Bu^{-q} \end{aligned}$$

and

$$\begin{aligned} u_t=\Delta _f u+Ae^{pu}+Be^{-pu}+D \end{aligned}$$

are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schrödinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Viêt Nam National University, Hanoi, Vietnam

    Nguyen Thac Dung, Nguyen Ngoc Khanh & Quốc Anh Ngô

  2. Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

    Quốc Anh Ngô

Authors
  1. Nguyen Thac Dung
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  2. Nguyen Ngoc Khanh
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  3. Quốc Anh Ngô
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Correspondence to Nguyen Thac Dung.

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Dung, N.T., Khanh, N.N. & Ngô, Q.A. Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces. manuscripta math. 155, 471–501 (2018). https://doi.org/10.1007/s00229-017-0946-3

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  • DOI: https://doi.org/10.1007/s00229-017-0946-3

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