Bulletin of the
Korean Mathematical Society BKMS

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