  In this paper, we consider a manifold evolving by a general geometric flow
and study parabolic equation \[ (\Delta
-q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We
establish space-time gradient estimates for positive solutions and elliptic
type gradient estimates for bounded positive solutions of this equation. By
integrating the gradient estimates, we derive the corresponding Harnack
inequalities. Finally, as applications, we give gradient estimates of some
specific parabolic equations.
