  Let $(M^{n},g,e^{-\phi}dv)$ be a weighted Riemannian manifold evolving by
geometric flow $\frac{\partial g}{\partial t}=2h(t),\,\,\,\frac{\partial
\phi}{\partial t}=\Delta \phi$. In this paper, we obtain a series of space-time
gradient estimates for positive solutions of a parabolic partial equation
$$(\Delta_{\phi}-\partial_{t})u(x,t)=q(x,t)u^{a+1}(x,t)+p(x,t)A(u(x,t))),\,\,\,\,(x,t)\in
M\times[0,T]$$ on a weighted Riemannian manifold under geometric flow. By
integrating the gradient estimates, we find the corresponding Harnack
inequalities.
