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Gradient estimates and Harnack inequalities of a nonlinear parabolic equation for the V-Laplacian

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Abstract

In this paper, we consider gradient estimates for the positive solutions to the following nonlinear parabolic equation:

$$\begin{aligned} u_t=\Delta _V u + au \log u \end{aligned}$$

on \(M \times [0, T]\), where a is a real constant. We obtain the Li-Yau type bounds of the above equation, which cover the estimates in Davies (Heat kernels and spectral theory 1989), Huang et al. (Ann Glob Anal Geom 43:209–232, 2013), Li and Xu (Adv Math 226:4456–4491, 2011) and Qian (J Math Anal Appl 409:556–566, 2014). Besides, as a corollary, we give a gradient estimate for the corresponding elliptic case:

$$\begin{aligned} \Delta _V u + au \log u = 0, \end{aligned}$$

which improves the estimates in Chen and Chen (Ann Glob Anal Geom 35:397–404, 2009) and Yang ( Proc AMS 136(11):4095–4102, 2008).

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Acknowledgments

This work is partially supported by NSFC, SRFDPHE and CSC of China. The authors would like to thank the referee for valuable suggestions which improved the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Qun Chen & Hongbing Qiu

  2. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103, Leipzig, Germany

    Hongbing Qiu

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  1. Qun Chen
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  2. Hongbing Qiu
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Correspondence to Hongbing Qiu.

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Chen, Q., Qiu, H. Gradient estimates and Harnack inequalities of a nonlinear parabolic equation for the V-Laplacian. Ann Glob Anal Geom 50, 47–64 (2016). https://doi.org/10.1007/s10455-016-9501-9

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  • DOI: https://doi.org/10.1007/s10455-016-9501-9

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