Archiv der Mathematik

, Volume 105, Issue 5, pp 479–490 | Cite as

Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds

  • Nguyen Thac Dung
  • Nguyen Ngoc Khanh


The purpose of this paper is to study gradient estimates of Hamilton–Souplet–Zhang type for the following general heat equation
$$u_t=\Delta_V u + au\log u+bu$$
on noncompact Riemannian manifolds. As its application, we show a Harnack inequality for the positive solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extension and improvement of the work of Souplet and Zhang (Bull London Math Soc 38:1045–1053, 2006), Ruan (Bull London Math Soc 39:982–988, 2007), Li (Nonlinear Anal 113:1–32, 2015), Huang and Ma (Gradient estimates and Liouville type theorems for a nonlinear elliptic equation, Preprint, 2015), and Wu (Math Zeits 280:451–468, 2015).


Gradient estimates General heat equation Laplacian comparison theorem V-Bochner–Weitzenböck Bakry–Emery Ricci curvature 

Mathematics Subject Classification

Primary 58J35 Secondary 35B53 35K0 


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  1. 1.
    Brighton K.: A Liouville-type theorem for smooth metric measure spaces. Jour. Geom. Anal. 23, 562–570 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. Calabi, An extension of E.Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. Jour., 25 (157), 45–56Google Scholar
  3. 3.
    Chen Q., Jost J., Qiu H. B.: Existence and Liouville theorems for V-harmonic maps from complete manifolds. Ann. Glob. Anal. Geom. 42, 565–584 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    N. T. Dung, Hamilton type gradient estimate for a nonlinear diffusion equation on smooth metric measure spaces, ManuscriptGoogle Scholar
  5. 5.
    Hamilton R. S.: A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1, 113–126 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    G. Y. Huang and B. Q. Ma, Gradient estimates and Liouville type theorems for a nonlinear elliptic equation (2015, preprint). arXiv:1505.01897v1
  7. 7.
    Li Y.: Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature. Nonlinear Anal. 113, 1–32 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li P., Yau S. T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 152–201 (1986)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li X. D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. Jour. Math. Pure. Appl. 84, 1295–1361 (2005)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ruan Q. H.: Elliptic-type gradient estimates for Schrödinger equations on noncompact manifolds. Bull. London Math. Soc. 39, 982–988 (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Souplet P., Zhang Q. S.: Sharp grandient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38, 1045–1053 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wu J. Y.: Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds. Jour. Math. Anal. Appl. 369, 400–407 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Wu J. Y.: Elliptic gradient estimates for a weighted heat equation and applications. Math. Zeits. 280, 451–468 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Mechanics and Informatics (MIM)Hanoi University of Sciences (HUS-VNU)Thanh XuanVietnam

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