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

Abstract

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.