# HARNACK INEQUALITY FOR A NONLINEAR PARABOLIC EQUATION UNDER GEOMETRIC FLOW

• Zhao, Liang (Department of Mathematics Nanjing University of Aeronautics and Astronautics)
• Published : 2013.09.30

#### Abstract

In this paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation $$\frac{{\partial}u}{{\partial}t}={\triangle}u-b(x,t)u^{\sigma}$$ under general geometric flow on complete noncompact manifolds, where 0 < ${\sigma}$ < 1 is a real constant and $b(x,t)$ is a function which is $C^2$ in the $x$-variable and $C^1$ in the$t$-variable. As an application, we get an interesting Harnack inequality.

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