Hamilton's gradient estimates and Liouville theorems for $u_{t}=\Delta u^{m}+au\log u+bu$ on Riemannian manifolds

Abstract

In this paper, we apply Nash-Moser iteration and Saloff-Coste's Sobolev inequalities to derive gradient estimates for positive solutions to a class of nonlinear parabolic equations of the form \begin{equation*} u_{t}=\Delta u^{m}+a(x, t)u\log u+b(x, t)u, \quad 1-\frac{2}{n}<m<1 \quad \text{or}\quad 1<m<1+\frac{1}{1+\sqrt{2n}}, \end{equation*} on a complete Riemannian manifold $(M,g)$ of dimension $n$, where $a(x, t)$, $b(x, t)$ are $C^{1}$ functions. For $a=b=0$, this equation becomes the porous medium equation (PME) or the fast diffusion equation (FDE) when $m>1$ or $m<1$, respectively. As consequences, we obtain Liouville type theorems for the corresponding constant coefficient equations when \begin{equation*} 1-\frac{1}{\sqrt{n-1}}<m<1\quad \text{and}\quad 1<m<1+\frac{1}{\sqrt{n-1}}. \end{equation*}