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Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 271–291 | Cite as

Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn–Hilliard systems

  • Takeshi Fukao
  • Taishi Motoda
Article

Abstract

A nonlinear diffusion equation with the Robin boundary condition is the main focus of this paper. The degenerate parabolic equations, such as the Stefan problem, the Hele-Shaw problem, the porous medium equation and the fast diffusion equation, are included in this class. By characterizing this class of equations as an asymptotic limit of the Cahn–Hilliard systems, the growth condition of the nonlinear term can be improved. In this paper, the existence and uniqueness of the solution are proved. From the physical view point, it is natural that, the Cahn–Hilliard system is treated under the homogeneous Neumann boundary condition. Therefore, the Cahn–Hilliard system subject to the Robin boundary condition looks like pointless. However, at some level of approximation, it makes sense to characterize the nonlinear diffusion equations.

Keywords

Cahn–Hilliard system Degenerate parabolic equation Robin boundary condition Growth condition 

Mathematics Subject Classification

35K61 35K65 35K25 35D30 80A22 

Notes

Acknowledgements

The first author is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Grant numbers 17K05321. Authors are deeply grateful to the anonymous referees for reviewing the original manuscript and for many valuable comments that helped to clarify and refine this paper. Authors also thank Richard Haase, Ph.D, from Edanz Group (http://www.edanzediting.com/ac) for editing a draft of this manuscript.

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Copyright information

© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationKyoto University of EducationKyotoJapan
  2. 2.Graduate School of EducationKyoto University of EducationKyotoJapan

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