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Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

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Abstract

Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation

$ \partial _t u = \Delta u + au\log u + qu $

on M × (0,∞), where a is a constant and q is a C 2 function. This result can be compared with the ones of Ma (JFA, 241, 374–382 (2006)) and Yang (PAMS, 136, 4095–4102 (2008)). Also, we obtain Hamilton’s gradient estimate for the Schödinger equation. This can be compared with the result of Ruan (JGP, 58, 962–966 (2008)).

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References

  1. Ma, L.: Gradient estimates for a simple elliptic equation on non-compact Riemannian manifolds. J. Funct. Anal., 241, 374–382 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Hamiltom, R. S.: A matrix harnack estimate for the heat equation. Comm. Anal. Geom., 1, 113–126 (1993)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Li, J.: 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Bakry, D., Ledoux, M.: A logarithmic Sobolev form of the Li-Yau parabolic inequality. Revista Mat. Iberoamericana, 22, 683–702 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Bakry, D., Baudoin, F., Bonnefont, M., et al.: Subelliptic Li-Yau estimates on three dimensional model spaces. In: Potential Theory and Stochastics in Albac: Aurel Cornea Memorial Volume, 2009

  10. Negrin, E.: Gradient estimates and a Liouville type theorem for the Schrödinger operator. J. Funct. Anal., 127, 198–203 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bakry, D.: On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. In: New trends in stochastic analysis (Charingworth, 1994), Taniguchi symposium, World Sci. Publ. River Edge, NJ, 1997, 43–75

    Google Scholar 

  12. Calabi, E.: An extension of E. Hopf’s maximum principle with application to Riemannian geometry. Duke Math. J., 25, 45–56 (1958)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Changshu Institute of Technology, Changshu, 215500, P. R. China

    Bin Qian

  2. Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS 5219, Toulouse, France

    Bin Qian

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  1. Bin Qian
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Correspondence to Bin Qian.

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Supported by the China Scholarship Council of China (Grant No. 2007U13020) during the 2007–2009 Academic Year

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Qian, B. Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Acta. Math. Sin.-English Ser. 27, 1071–1078 (2011). https://doi.org/10.1007/s10114-011-8474-7

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  • DOI: https://doi.org/10.1007/s10114-011-8474-7

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