Harnack inequality for a nonlinear parabolic equation under geometric flow
Bull. Korean Math. Soc. 2013 Vol. 50, No. 5, 1587-1598
Printed September 1, 2013
Liang Zhao
Nanjing University of Aeronautics and Astronautics
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.
Keywords : parabolic equation, positive solutions, geometric flow, Harnack inequality
MSC numbers : 53C44
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