Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Yang, Yunyan

doi:10.1090/S0002-9939-08-09398-2

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds
HTML articles powered by AMS MathViewer

by Yunyan Yang

Proc. Amer. Math. Soc. 136 (2008), 4095-4102

DOI: https://doi.org/10.1090/S0002-9939-08-09398-2

Published electronically: June 11, 2008

PDF | Request permission

Abstract:

Let $(M,g)$ be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation \[ \frac {\partial u}{\partial t}=\Delta u+au\log u+bu\] on $M\times [0,+\infty )$, where $a$, $b$ are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).

References

Saïd Asserda, Un théorème de Liouville pour l’opérateur de Schrödinger avec dérive, C. R. Math. Acad. Sci. Paris 342 (2006), no. 6, 393–398 (French, with English and French summaries). MR 2209217, DOI 10.1016/j.crma.2006.01.007
Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56. MR 92069, DOI 10.1215/S0012-7094-58-02505-5
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. MR 385749, DOI 10.1002/cpa.3160280303
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Jiayu 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. MR 1125225, DOI 10.1016/0022-1236(91)90110-Q
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
Li Ma, Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds, J. Funct. Anal. 241 (2006), no. 1, 374–382. MR 2264255, DOI 10.1016/j.jfa.2006.06.006
Antonios D. Melas, A Liouville type theorem for the Schrödinger operator, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3353–3359. MR 1623036, DOI 10.1090/S0002-9939-99-05026-1
E. R. Negrín, Gradient estimates and a Liouville type theorem for the Schrödinger operator, J. Funct. Anal. 127 (1995), no. 1, 198–203. MR 1308622, DOI 10.1006/jfan.1995.1008
Peter Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note Series, vol. 325, Cambridge University Press, Cambridge, 2006. MR 2265040, DOI 10.1017/CBO9780511721465
Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203