Hamilton's gradient estimates and Liouville theorems for $u_{t}=Δu^{m}+au\log u+bu$ 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}=Δu^{m}+a(x, t)u\log u+b(x, t)u, \quad 1-\frac{2}{n} 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*}