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

Authors:

Li Chen Wenyi Chen

Institution:

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

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.

Keywords:

Nonlinear parabolic equation Gradient estimate Positive solution

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