Gradient estimates of nonlinear equation on complete noncompact metric measure space with compact boundary

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.