Skip to main content
SpringerLink
Log in

Harnack differential inequalities for the parabolic equation u t = ℒF(u) on Riemannian manifolds and applications

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, let (Mn, g) be an n-dimensional complete Riemannian manifold with the m-dimensional Bakry–Émery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation

$${u_t} = LF\left( u \right) = \Delta F\left( u \right) - \nabla f \cdot \nabla F\left( u \right),$$

on compact Riemannian manifolds Mn, where F ∈ C 2(0,∞), F′ > 0 and f is a C 2-smooth function defined on Mn. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bakry, D., Emery, M.: Diffusion Hypercontractivites, Sminaire de Probabilits XIX, 1983/1984, Lect. Notes in Math., 1123, Springer-Verlag, Berlin, 1985, 177–206

    Google Scholar 

  2. Bakry, D.: Lhypercontractivité et son utilisation en théorie des semigroupes, Lecture Notes in Math., vol. 1581, Springer-Verlag, Berlin/New York, 1994, 1–114

    Google Scholar 

  3. Bakry, D., Qian, Z. M.: Volume comparison theorems without Jacobi fields, in: Current Trends in Potential Theory, in: Theta Ser. Adv. Math., vol. 4, Theta, Bucharest, 115–122 (2005)

    Google Scholar 

  4. Cheng, S. Y., Yau, S. T.: Defferential equations on Riemannnian manifolds and their geometric applications. Comm. Pure Appl. Math., 28, 333–354 (1975)

    Article  MathSciNet  Google Scholar 

  5. Hamilton, R. S.: A matrix Harnack estimates for the heat equation. Comm. Anal. Geom., 1, 113–126 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Herrero, M. A., Pierre, M.: The Cauchy problem for u t = Δu m when 0 < m < 1. Trans. Amer. Math. Soc., 291, 145–158 (1985)

    MathSciNet  MATH  Google Scholar 

  7. Huang, G., Huang, Z., Li, H.: Gradient estimates for the porous medium equations on Riemannian manifolds. J. Geom. Anal., 23, 1851–1875 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, G.: Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian. Pacific Journal of Mathematics, 268(1), 47–78 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, J. Y.: Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal., 100, 233–256 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, J., Xu, X.: Defferential Harnack inequalities on Riemannian manifolds I: Linear heat equation. Adv. Math., 226, 4456–4491 (2001)

    Article  MATH  Google Scholar 

  11. Li, X. D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl., 84, 1295–1361 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, P., Yau, S. T.: On the parabolic kernel of the Schrödinger operator. Acta Math., 156, 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  13. Lu, P., Ni, L., V´azquez, J. L., et al.: Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds. J. Math. Pures Appl., 91, 1–19 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ma, L., Zhao, L., Song, X. F.: Gradient estimate for the degenerate parabolic equation u t = ΔF(u)+H(u) on manifolds. J. Differential Equations, 224, 1157–1177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ruan, Q. H.: Gradient estimate for Schrödinger operators on manifolds. J. Geom. Phys., 58, 962–966 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Souplet, P., Zhang, Q. S.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc., 38, 1045–1053 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wu, J. Y.: Li–Yau type estimates for a nonlinear parabolic equation on complete manifolds. J. Math. Anal. Appl., 369, 400–407 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, J. Y.: Gradient estimates for a nonlinear diffusion equation on complete manifolds. J. Part. Diff. Eq., 23(1) 68–79

  19. Xu, X.: Gradient estimates for u t = ΔF(u) on manifolds and some Liouville-type theorems. J. Differential Equations, 252, 1403–1420 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yau, S. T.: On the Harnack inequalities of partial differential equations. Comm. Anal. Geom., 2(3), 421–451 (1994)

    Article  MathSciNet  Google Scholar 

  21. Yang, Y. Y.: Gradient estimate for a nonlinear parabolic equation on Riemannian manifold. Proc. Amer. Math. Soc., 136, 4095–4102 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhu, X. B.: Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds. Proc. Amer. Math. Soc., 139(5), 1637–1644 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhu, X. B.: Hamilton’s gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannnian manifolds. J. Math. Anal. Appl., 402, 201–206 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hefei Normal University, Hefei, 230601, P. R. China

    Wen Wang

  2. School of Mathematical Science, University of Science and Technology of China, Hefei, 230026, P. R. China

    Wen Wang

Authors
  1. Wen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen Wang.

Additional information

Supported by Universities Natural Science Foundation of Anhui Province (Grant No. KJ2016A310)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, W. Harnack differential inequalities for the parabolic equation u t = ℒF(u) on Riemannian manifolds and applications. Acta. Math. Sin.-English Ser. 33, 620–634 (2017). https://doi.org/10.1007/s10114-016-6260-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-016-6260-2

Keywords

MR(2010) Subject Classification

Search

Navigation