Taiwanese Journal of Mathematics

Gradient Estimates for the Nonlinear Parabolic Equation with Two Exponents on Riemannian Manifolds

Songbo Hou

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access


In this paper, we study the nonlinear parabolic equation with two exponents on complete noncompact Riemannian maniflods. The special types of such equation include the Fisher-KPP equation, the parabolic Allen-Cahn equation and the Newell-Whitehead equation. We get the Souplet-Zhang's gradient estimates for the positive solutions to such equation. We also obtain the Liouville theorem for positive ancient solutions. Our results extend those of Souplet-Zhang (Bull. London. Math. Soc. 38 (2006), 1045--1053) and Zhu (Acta Math. Sci. Ser. B 36 (2016), no. 2, 514--526).

Article information

Taiwanese J. Math., Advance publication (2020), 10 pages.

First available in Project Euclid: 14 April 2020

Permanent link to this document

Digital Object Identifier

Primary: 35K55: Nonlinear parabolic equations 58J35: Heat and other parabolic equation methods

gradient estimate nonlinear parabolic equation Liouville theorem


Hou, Songbo. Gradient Estimates for the Nonlinear Parabolic Equation with Two Exponents on Riemannian Manifolds. Taiwanese J. Math., advance publication, 14 April 2020. doi:10.11650/tjm/200401. https://projecteuclid.org/euclid.twjm/1586851304

Export citation


  • M. Băileşteanu, A Harnack inequality for the parabolic Allen-Cahn equation, Ann. Global Anal. Geom. 51 (2017), no. 4, 367–378.
  • D. Booth, J. Burkart, X. Cao, M. Hallgren, Z. Munro, J. Snyder and T. Stone, A differential Harnack inequality for the Newell-Whitehead equation, arXiv:1712.04024.
  • E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56.
  • X. Cao, B. Liu, I. Pendleton and A. Ward, Differential Harnack estimates for Fisher's equation, Pacific J. Math. 290 (2017), no. 2, 273–300.
  • S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354.
  • R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), no. 4, 355–369.
  • X. Geng and S. Hou, Gradient estimates for the Fisher-KPP equation on Riemannian manifolds, Bound. Value Probl. 2018 (2018), no. 25, 12 pp.
  • R. S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113–126.
  • S. Hou, Gradient estimates for the Allen-Cahn equation on Riemannian manifolds, Proc. Amer. Math. Soc. 147 (2019), no. 2, 619–628.
  • G. Huang and B. Ma, Hamilton-Souplet-Zhang's gradient estimates for two types of nonlinear parabolic equations under the Ricci flow, J. Funct. Spaces 2016 (2016), Art. ID 2894207, 7 pp.
  • A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Moscou Sér. Internat. A 1 (1937), 1–25; English transl. in: P. Pelcé, Dynamics of Curved Fronts, 105–130, Perspectives in Physics, Academic Press, 1988.
  • J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal. 100 (1991), no. 2, 233–256.
  • P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201.
  • L. Ma, Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds, J. Funct. Anal. 241 (2006), no. 1, 374–382.
  • E. R. Negrín, Gradient estimates and a Liouville type theorem for the Schrödinger operator, J. Funct. Anal. 127 (1995), no. 1, 198–203.
  • A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (1969), no. 2, 279–303.
  • 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.
  • Y. Yang, Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4095–4102.
  • S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228.
  • X. Zhu, Gradient estimates and Liouville theorems for linear and nonlinear parabolic equations on Riemannian manifolds, Acta Math. Sci. Ser. B (Engl. Ed.) 36 (2016), no. 2, 514–526.