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Gradient estimates for a nonlinear diffusion equation on complete manifolds

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Abstract

This paper deals with the gradient estimates of the Hamilton type for the positive solutions to the following nonlinear diffusion equation:

$${u_t} = \Delta u + \nabla \phi \cdot \nabla u + a\left( x \right)u\ln u + b\left( x \right)u$$

on a complete noncompact Riemannian manifold with a Bakry-Emery Ricci curvature bounded below by −K (K ≥ 0), where φ is a C 2 function, a(x) and b(x) are C 1 functions with certain conditions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing, 210044, China

    Jiaxian Wu

  2. Department of Mathematics, Putian University, Putian, Fujian, 351100, China

    Qihua Ruan

  3. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yihu Yang

Authors
  1. Jiaxian Wu
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  2. Qihua Ruan
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  3. Yihu Yang
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Corresponding author

Correspondence to Jiaxian Wu.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11171253, 11471175), the Fujian Provincial National Natural Science Foundation of China (No. 2012J01015) and the Startup Foundation for Introducing Talent of Nuist(No. 2014r030) and the Pre-research Foundation of NSFC(No. 2014x025).

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Wu, J., Ruan, Q. & Yang, Y. Gradient estimates for a nonlinear diffusion equation on complete manifolds. Chin. Ann. Math. Ser. B 36, 1011–1018 (2015). https://doi.org/10.1007/s11401-015-0922-8

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  • DOI: https://doi.org/10.1007/s11401-015-0922-8

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