Harnack Inequalities for Simple Heat Equations on Riemannian Manifolds

  • Li MaEmail author
  • Yang Liu


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 Wi and V only depend on the space variable xM. 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).


Positive solution Nonlinear heat equation Gradient estimate Harnack inequality 

Mathematics Subject Classification (2010)

53C44 53C21 35K05 



The authors would like to thank the unknown referees for a helpful suggestion.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China

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