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Annals of Global Analysis and Geometry

, Volume 50, Issue 1, pp 47–64 | Cite as

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

  • Qun Chen
  • Hongbing QiuEmail author
Article

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).

Keywords

Gradient estimate Nonlinear parabolic equation Positive solution Harnack inequality 

Mathematics Subject Classification

35B45 35K55 

Notes

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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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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