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

TL;DR

This paper establishes gradient estimates for porous medium and fast diffusion equations on complete noncompact metric measure spaces with boundary, extending classical results to more general geometric settings.

Contribution

It derives Li-Yau and Souplet-Zhang type gradient estimates for nonlinear diffusion equations on noncompact metric measure spaces with boundary.

Findings

Li-Yau gradient estimates derived for the equations

Souplet-Zhang type estimates established

Local gradient estimates for related elliptic equations obtained

Abstract

In the paper, we derive Li-Yau gradient estimates and Souplet Zhang type estimates of the following equation \begin{equation*} \begin{split} u_t= \Delta_\xi p+\lambda u+A(u) , \end{split} \end{equation*} on complete noncompact 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 complete noncompact manifold with compact boundary.