• Dung, Ha Tuan
  • Received : 2017.01.27
  • Accepted : 2018.08.29
  • Published : 2018.11.01


This note is motivated by gradient estimates of Li-Yau, Hamilton, and Souplet-Zhang for heat equations. In this paper, our aim is to investigate Yamabe equations and a non linear heat equation arising from gradient Ricci soliton. We will apply Bochner technique and maximal principle to derive gradient estimates of the general non-linear heat equation on Riemannian manifolds. As their consequence, we give several applications to study heat equation and Yamabe equation such as Harnack type inequalities, gradient estimates, Liouville type results.


gradient estimates;Bakry-${\acute{E}}mery$ curvature;Bochner's technique;Harnack-type inequalities;Liouville-type theorems


  1. H. T. Dung, Gradient estimates and Harnack inequalities for Yamabe-type parabolic equations on Riemannian manifolds, Differential Geom. Appl. 60 (2018), 39-48.
  2. N. T. Dung and N. N. Khanh, Gradient estimates of Hamilton-Souplet-Zhang type for a general heat equation on Riemannian manifolds, Arch. Math. (Basel) 105 (2015), no. 5, 479-490.
  3. R. S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113-126.
  4. P. Li and S.-T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986), no. 3-4, 153-201.
  5. Y. Li, Li-Yau-Hamilton estimates and Bakry-Emery-Ricci curvature, Nonlinear Anal. 113 (2015), 1-32.
  6. L. Ma, Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds, J. Funct. Anal. 241 (2006), no. 1, 374-382.
  7. P. Mastrolia, M. Rigoli, and A. G. Setti, Yamabe-type equations on complete, noncompact manifolds, Progress in Mathematics, 302, Birkhauser/Springer Basel AG, Basel, 2012.
  8. H. Qiu, The heat flow of V -harmonic maps from complete manifolds into regular balls, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2271-2280.
  9. Q. Ruan, Elliptic-type gradient estimate for Schrodinger equations on noncompact manifolds, Bull. Lond. Math. Soc. 39 (2007), no. 6, 982-988.
  10. P. Souplet and Q. S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc. 38 (2006), no. 6, 1045-1053.
  11. J.-Y. Wu, Elliptic gradient estimates for a weighted heat equation and applications, Math. Z. 280 (2015), no. 1-2, 451-468.
  12. M.-F. Bidaut-Veron and L. Veron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489-539.
  13. L. Brandolini, M. Rigoli, and A. G. Setti, Positive solutions of Yamabe type equations on complete manifolds and applications, J. Funct. Anal. 160 (1998), no. 1, 176-222.
  14. E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45-56.
  15. Q. Chen, J. Jost, and H. Qiu, Existence and Liouville theorems for V-harmonic maps from complete manifolds, Ann. Global Anal. Geom. 42 (2012), no. 4, 565-584.
  16. Q. Chen, J. Jost, and H. Qiu, Omori-Yau maximum principles, V-harmonic maps and their geometric applications, Ann. Global Anal. Geom. 46 (2014), no. 3, 259-279.
  17. Q. Chen, J. Jost, and G. Wang, A maximum principle for generalizations of harmonic maps in Hermitian, affine, Weyl, and Finsler geometry, J. Geom. Anal. 25 (2015), no. 4, 2407-2426.
  18. Q. Chen and H. Qiu, Gradient estimates and Harnack inequalities of a nonlinear parabolic equation for the V-Laplacian, Ann. Global Anal. Geom. 50 (2016), no. 1, 47-64.
  19. Q. Chen and H. Qiu, Rigidity of self-shrinkers and translating solitons of mean curvature flows, Adv. Math. 294 (2016), 517-531.


Supported by : Hanoi Pedagogical University No. 2