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In the paper, we derive Li-Yau gradient estimates and Souplet Zhang typeestimates of the following equation \begin{equation*} \begin{split} u_t=\Delta_\xi p+\lambda u+A(u) , \end{split} \end{equation*} on completenoncompact metric measure space $ (M, g,e^{-\xi}dv_g) $ with compact boundary.We will also give the local gradient estimates of the equation\begin{equation*} \Delta_\xi(u^p)+\lambda u+A(u)=0, \end{equation*} on completenoncompact manifold with compact boundary.

Gradient estimates for the porous medium type equations and fast  diffusion type equations on complete noncompact metric measure space with  compact boundary