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Volume 23, Issue 1
Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds

Jiayong Wu

DOI: 10.4208/jpde.v23.n1.4

J. Part. Diff. Eq., 23 (2010), pp. 68-79.

Published online: 2010-02

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  • Abstract

Let (M,g) be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation u_t=Δu-∇φ·∇u-au\log u-bu, where φ is a C^2 function, and a ≠ 0 and b are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).

  • Keywords

Local gradient estimate nonlinear diffusion equation Bakry-Émery Ricci curvature

  • AMS Subject Headings

58J35 58J35 58J05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-23-68, author = {Jiayong Wu }, title = {Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds}, journal = {Journal of Partial Differential Equations}, year = {2010}, volume = {23}, number = {1}, pages = {68--79}, abstract = {

Let (M,g) be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation u_t=Δu-∇φ·∇u-au\log u-bu, where φ is a C^2 function, and a ≠ 0 and b are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v23.n1.4}, url = {http://global-sci.org/intro/article_detail/jpde/5222.html} }
TY - JOUR T1 - Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds AU - Jiayong Wu JO - Journal of Partial Differential Equations VL - 1 SP - 68 EP - 79 PY - 2010 DA - 2010/02 SN - 23 DO - http://doi.org/10.4208/jpde.v23.n1.4 UR - https://global-sci.org/intro/article_detail/jpde/5222.html KW - Local gradient estimate KW - nonlinear diffusion equation KW - Bakry-Émery Ricci curvature AB -

Let (M,g) be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation u_t=Δu-∇φ·∇u-au\log u-bu, where φ is a C^2 function, and a ≠ 0 and b are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).

Jiayong Wu . (2010). Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds. Journal of Partial Differential Equations. 23 (1). 68-79. doi:10.4208/jpde.v23.n1.4
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