Abstract
In this paper, we derive Li-Yau gradient estimates for the positive solution of a nonlinear parabolic equation \(u_{t}={\Delta } u -qu-au(\ln u)^{\alpha }\), where q is a C 2 function and a,α are constants, on a complete manifold (M,g) with bounded below Ricci curvature. The results generalize classical Li-Yau gradient estimates and some recent works on this direction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Calabi, E.: An extension of E. Hopf’s maximum principle with application to Riemannian geometry. Duck Math. J. 25, 45–46 (1958). MR0092069 (19, 1056e)
Chen, L., Chen, W.: Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Ann. Glob. Anal. Geom. 35 (4), 397–404 (2009). MR2506242 (2010k: 35501)
Cheng, S.-Y., Yau, S.-T. : Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure. Appl. Math. 28 (3), 333–354 (1975). MR0385749 (52 # 6608)
Hamilton, R.S.: The Harnack estimate for the Ricci flow. J. Differential Geom. 37 (1), 225–243 (1993). MR1198607 (93k: 58052)
Hsu, S.: Gradient estimates for a nonlinear parabolic equation under Ricci flow. Differ. Integr. Equ. 24 (7–8), 645–652 (2011). MR2830313 (2012h: 53150)
Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (3–4), 153–201 (1986). MR0834612 (87f: 58156)
Li, Y., Zhu, X.: Preparation
Ma, L.: Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds. J. Funct. Anal. 241 (1), 374–382 (2006). MR2264255 (2007e: 53034)
Peleman, G.: The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/021159
Qian, B.: Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Acta. Math. Sin. (Engl. Ser.) 27 (6), 1071–1078 (2011). MR2795355 (2012c: 35438)
Schoen, R., Yau, S.-T.: Lectures on differential geometry, Conference Proceedings and Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu. Translated from the Chinese by Ding and S. Y. Cheng. Preface translated from the Chinese by Kaising Tso. iN: Conference Proceedings and Lecture Notes in Geometry and Topology, I, pp. v+235 International Press, Cambridge, MA. ISBN: 1-57146-012-8 MR1333601 (97d: 53001) (1994)
Wu, J.-Y.: Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds. J. Math. Anal. Appl. 369 (1), 400–407 (2010). MR2643878 (2011b: 35432)
Yang, Y.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc. Amer. Math. Soc. 136 (11), 4095–4102 (2008). MR2425752 (2009d: 58048)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, X., Li, Y. Li-Yau Estimates for a Nonlinear Parabolic Equation on Manifolds. Math Phys Anal Geom 17, 273–288 (2014). https://doi.org/10.1007/s11040-014-9155-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11040-014-9155-4