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Gradient estimates for a nonlinear parabolic equation and Liouville theorems

  • Jia-Yong Wu
Article
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Abstract

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation’s coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.

Mathematics Subject Classification

Primary 53C21 58J35 Secondary 35B53 35K55 

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Notes

Acknowledgements

The author thanks Professor Jeffrey S. Case for helpful discussions. The author also thanks the referee for making valuable comments and suggestions and pointing out many errors which helped to improve the exposition of the paper. This work is supported by the NSFC (11671141) and the Natural Science Foundation of Shanghai (17ZR1412800).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Maritime UniversityShanghaiPeople’s Republic of China

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