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The Harnack estimate for a nonlinear parabolic equation under the Ricci flow

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Abstract

Let (M,g(t)), 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci curvature, |Rc| ≤ C/t for some constant C > 0 and g(t) evolving by the Ricci flow

$\frac{{\partial g_{ij} }} {{\partial t}} = - 2R_{ij} . $

. In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type

$u_t = \Delta u - au\log u - bu$

on M × (0, T], where a > 0 and b ∈ ℝ.

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Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, East Campus, China Agricultural University, Beijing, 100083, P. R. China

    Song Bo Hou

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  1. Song Bo Hou
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Correspondence to Song Bo Hou.

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Supported by National Natural Science Foundation of China (Grant Nos. 10926109 and 11001268) and Chinese Universities Scientific Fund (2009JS32 and 2009-2-05)

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Hou, S.B. The Harnack estimate for a nonlinear parabolic equation under the Ricci flow. Acta. Math. Sin.-English Ser. 27, 1935–1940 (2011). https://doi.org/10.1007/s10114-011-0074-z

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  • DOI: https://doi.org/10.1007/s10114-011-0074-z

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