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Differential Harnack inequality for the nonlinear heat equations. (English) Zbl 1287.53062

Summary: In this paper, we establish some differential Harnack inequalities for positive solutions to the nonlinear heat equations with potentials evolving by the Bernhard List’s flow. Our theorems generalize X. Cao and Z. Zhang’s results [Springer Proc. Math. 8, 87–98 (2011; Zbl 1228.53078)].

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K05 Heat equation

Citations:

Zbl 1228.53078
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References:

[1] X. Cao and Z. Zhang, Differential Harnack estimates for parabolic equations, in Complex and differential geometry , Springer Proc. Math., 8, Springer, Heidelberg, 2011, pp. 87-98. · Zbl 1228.53078 · doi:10.1007/978-3-642-20300-8_5
[2] H.-D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds, Math. Ann. 331 (2005), no. 4, 795-807. · Zbl 1083.58024 · doi:10.1007/s00208-004-0605-3
[3] X. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow, J. Funct. Anal. 255 (2008), no. 4, 1024-1038. · Zbl 1146.58014 · doi:10.1016/j.jfa.2008.05.009
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[6] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153-201. · Zbl 0611.58045 · doi:10.1007/BF02399203
[7] B. List, Evolution of an extended Ricci flow system, Comm. Anal. Geom. 16 (2008), no. 5, 1007-1048. · Zbl 1166.53044 · doi:10.4310/CAG.2008.v16.n5.a5
[8] S. Liu, Gradient estimates for solutions of the heat equation under Ricci flow, Pacific J. Math. 243 (2009), no. 1, 165-180. · Zbl 1180.58017 · doi:10.2140/pjm.2009.243.165
[9] S.-T. Yau, On the Harnack inequalities of partial differential equations, Comm. Anal. Geom. 2 (1994), no. 3, 431-450. · Zbl 0841.58059
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