Gradient estimates for porous medium and fast diffusion equations on Riemannian manifolds via Moser iteration
Shansong Huang, Bin Shen · Communications on Pure & Applied Analysis · 2025
Let $ (M^n,g) $ be a complete noncompact Riemannian manifold without boundary. In this article, we apply the Moser iteration technique to study the local gradient estimates of the positive solution to the porous medium and fast diffusion equations$ \begin{equation*} \Delta(u^m(x,t))-\partial_t u(x,t) = 0,\quad m>0, \end{equation*} $on $ M\times(0,\infty) $. There are several gradient estimates about the positive solutions to those equations. Our conclusions have better estimates compared to known results. In addition, the method we utilize in this article expands the application range of the Moser iteration in elliptic and parabolic equations. Moreover, the gradient estimates permit us to give some Harnack inequalities and Liouville-type theorems.