Harnack Inequalities for Simple Heat Equations on Riemannian Manifolds

Abstract

In this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem

$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$

on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature:

$ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $

where W_i and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).