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Science China Mathematics

, Volume 62, Issue 5, pp 839–852 | Cite as

The Cauchy problem for a class of nonlinear degenerate parabolic-hyperbolic equations

  • Hua ChenEmail author
  • Jinpeng Zhan
  • Xin Hu
Articles Progress of Projects Supported by NSFC
  • 72 Downloads

Abstract

In this article, we prove a general existence theorem for a class of nonlinear degenerate parabolichyperbolic equations. Since the regions of parabolicity and hyperbolicity are coupled in a way that depends on the solution itself, there is almost no hope of decoupling the regions and then taking into account the parabolic and the hyperbolic features separately. The existence of solutions can be obtained by finding the limit of solutions for the regularized equation of strictly parabolic type. We use the energy methods and vanishing viscosity methods to prove the local existence and uniqueness of solution.

Keywords

parabolic-hyperbolic equations energy methods vanishing viscosity methods 

MSC(2010)

35K55 35K65 35L60 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11631011 and 11626251). The authors are very grateful to the reviewers for their kind comments.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.School of Mathematics and Statistics and Computational Science Hubei Key LaboratoryWuhan UniversityWuhanChina

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