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

In the paper, we derive Li-Yau gradient estimates and Souplet Zhang type estimates of the following equation \begin{equation*} \begin{split} u_t= Δ_ξp+λu+A(u) , \end{split} \end{equation*} on complete noncompact metric measure space $(M,g,e^{-\xi }d{v}_g)$ with compact boundary. We will also give the local gradient estimates of the equation \begin{equation*} Δ_ξ(u^p)+λu+A(u)=0, \end{equation*} on complete noncompact manifold with compact boundary.