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Gradient estimates for a weighted nonlinear parabolic equation

Original Paper

Abstract

Let \((M^{n},g,e^{-f}dv)\) be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation
$$\begin{aligned} \left( \varDelta _{f}-\frac{\partial }{\partial t}\right) u(x,t)+q(x,t)u(x,t)+au(x,t)(\ln u(x,t))^{\alpha }=0, \end{aligned}$$
where \((x,t)\in M\times (-\infty ,\infty )\) and a, \(\alpha \) are arbitrary constants. Under the assumption that the \(\infty \)-Bakry-Émery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton’s type and Souplet–Zhang’s type) gradient estimates to positive smooth solutions of this equation.

Keywords

Gradient estimates Weighted nonlinear parabolic equation Bakry-Émery Ricci curvature 

Mathematics Subject Classification

35B45 53C44 

Notes

Acknowledgements

The author would like to thank the referee for his/her comments and suggestions.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsNorthwest Normal UniversityLanzhouPeople’s Republic of China

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