  In this paper, firstly, we study gradient estimates for positive solution of
the following equation \begin{equation*} \Delta_\xi(u)-\partial_t u- q u
=A(u),t\in (-\infty,\infty) \end{equation*} on metric measure space $
(M,g,e^{-\xi}\mathrm{d} v_g)$ with boundary , where $ \Delta_\xi=\Delta +\left
\langle \nabla\cdot , \nabla \xi\right \rangle $. For this equation, we derive
Li-Yau type gradient estimates and Hamilton's type gradient estimates.
Secondly, we obtain gradient estimates for positive solution of the following
elliptical type equation \begin{equation} \Delta_\xi(u)- q u
=A(u)\end{equation} on complete noncompact metric measure space
$(M,g,e^{-\xi}\mathrm{d} v_g)$ with boundary.
