## Differential and Integral Equations

### Gradient estimates for a nonlinear parabolic equation under Ricci flow

Shu-Yu Hsu

#### Abstract

Let $(M,g(t))$, $0\le t\le T$, be an n-dimensional complete noncompact manifold, $n\ge 2$, with bounded curvatures and metric $g(t)$ evolving by the Ricci flow $\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$. We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation $\frac{{\partial} u}{{\partial} t}=\Delta u-au\log u-qu,$ where $a\in\mathbb R$ is a constant and $q$ is a smooth function on $M\times [0,T]$.

#### Article information

Source
Differential Integral Equations, Volume 24, Number 7/8 (2011), 645-652.

Dates
First available in Project Euclid: 27 December 2012