Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds

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• 作者：Guangyue Huang (1)
  Zhijie Huang (2)
  Haizhong Li (2)
  
• 关键词：Gradient estimate ; Nonlinear parabolic equation ; Li ; Yau type estimate ; Harnack inequality ; Primary 35B45 ; Secondary 35K55
• 刊名：Annals of Global Analysis and Geometry
• 出版年：2013
• 出版时间：March 2013
• 年：2013
• 卷：43
• 期：3
• 页码：209-232
• 全文大小：272 KB
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• 作者单位：Guangyue Huang (1)
  Zhijie Huang (2)
  Haizhong Li (2)
  
  1. Department of Mathematics, Henan Normal University, Xinxiang, 453007, People’s Republic of China
  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
  
• ISSN：1572-9060

文摘

Let (M n , g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: $$u_t=\Delta u+au\log u+bu$$ on M n ?× [0,T], where a, b are two real constants. We derive local gradient estimates of the Li-Yau type for positive solutions of the above equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results extend the ones of Davies in Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol 92, Cambridge University Press, Cambridge,1989, and Li and Xu in Adv Math 226:4456-491 (2011).