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HARNACK INEQUALITY FOR A NONLINEAR PARABOLIC EQUATION UNDER GEOMETRIC FLOW
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 Title & Authors
HARNACK INEQUALITY FOR A NONLINEAR PARABOLIC EQUATION UNDER GEOMETRIC FLOW
Zhao, Liang;
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 Abstract
In this paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds, where 0 < < 1 is a real constant and is a function which is in the -variable and in the-variable. As an application, we get an interesting Harnack inequality.
 Keywords
parabolic equation;positive solutions;geometric flow;Harnack inequality;
 Language
English
 Cited by
 References
1.
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampre equations, Springer, New York, 1982.

2.
E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), no. 1, 45-56. crossref(new window)

3.
R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113-126.

4.
S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal. 255 (2008), no. 4, 1008-1023. crossref(new window)

5.
P. Li and S. T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986), no. 3-4, 153-201. crossref(new window)

6.
S. P. Liu, Gradient estimates for solutions of the heat equation under Ricci flow, Pacific J. Math. 243 (2009), no. 1, 165-180. crossref(new window)

7.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, (2002), arXiv math.DG/0211159.

8.
J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pacific J. Math. 253 (2011), no. 2, 489-510. crossref(new window)

9.
J. Zhang and B. Q. Ma, Gradient estimates for a nonlinear equation ${\Delta}_fu+cu^{-{\alpha}}$= 0 on complete noncompact manifolds, Commun. Math. 19 (2011), no. 1, 73-84.

10.
X. B. Zhu, Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds, Nonlinear Anal. 74 (2011), no. 15, 5141-5146. crossref(new window)