Harnack inequalities for a class of heat flows with nonlinear reaction terms

Abstract

A class of semilinear heat flows with general nonlinear reaction terms is considered on complete Riemannian manifolds with Ricci curvature bounded from below. Two types of (space-time and space only) gradient estimates are established for positive solutions to the flow, and the corresponding Harnack inequalities are obtained to allow for comparison of solutions. Some specific examples of the reaction term such as logarithmic reaction, Fisher-KPP and Allen-Cahn equations are discussed as applications of the estimates so derived. Referring to logarithmic nonlinearities, some discussions are made on Liouville type properties of bounded solutions.