Some gradient estimates for a diffusion equation on Riemannian manifolds

In this note we present some gradient estimates for the diffusion equation ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u-\mathrm{\nabla }\varphi \cdot \mathrm{\nabla }u$ on Riemannian manifolds, where $\varphi$ is a C^2 function, which generalize estimates of R. Hamilton’s and Qi S. Zhang’s on the heat equation.