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Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds

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Abstract

In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation

$$u_t=\Delta u+au\, {\rm log}\, u+bu\quad {\rm in}\,M$$

, where a, b are real constants, M is a complete noncompact Riemannian manifold. As a corollary, we give a local gradient estimate for the corresponding elliptic equation:

$$\Delta u+au\,{\rm log}\, u+bu=0\quad {\rm in}\,M$$

, which improves and extends the result of Ma (J Funct Anal 241:374–382, 2006) and get a bound for the positive solution to this elliptic equation.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Li Chen & Wenyi Chen

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  1. Li Chen
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  2. Wenyi Chen
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Correspondence to Wenyi Chen.

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Chen, L., Chen, W. Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Ann Glob Anal Geom 35, 397–404 (2009). https://doi.org/10.1007/s10455-008-9141-9

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  • DOI: https://doi.org/10.1007/s10455-008-9141-9

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