Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds

Abstract:

Let $M$ be a complete noncompact Riemannian manifold of dimension $n$. In this paper, we derive a local gradient estimate for positive solutions of fast diffusion equations \begin{align*} \partial _{t}u=\Delta u^{\alpha },\ \ 1-\frac {2}{n}<\alpha <1 \end{align*} on $M\times (-\infty ,0]$. We also obtain a theorem of Liouville type for positive solutions of the fast diffusion equation.