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


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.


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

Mathematics Subject Classification

35K61 35K65 35K25 35D30 80A22 



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 ( for editing a draft of this manuscript.


  1. 1.
    Aiki, T.: Two-phase Stefan problems with dynamic boundary conditions. Adv. Math. Sci. Appl. 2, 253–270 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aiki, T.: Multi-dimensional Stefan problems with dynamic boundary conditions. Appl. Anal. 56, 71–94 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aiki, T.: Periodic stability of solutions to some degenerate parabolic equations with dynamic boundary conditions. J. Math. Soc. Jpn. 48, 37–59 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Akagi, G.: Energy solutions of the Cauchy–Neumann problem for porous medium equations. In: Discrete and Continuous Dynamical Systems, supplement 2009, pp. 1–10. American Institute of Mathematical Sciences (2009)Google Scholar
  5. 5.
    Barbu, V.: Nonlinear differential equations of monotone types in Banach spaces. Springer, London (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Colli, P., Fukao, T.: Equation and dynamic boundary condition of Cahn–Hilliard type with singular potentials. Nonlinear Anal. 127, 413–433 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colli, P., Fukao, T.: Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems. J. Differ. Equ. 260, 6930–6959 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colli, P., Visintin, A.: On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ. 15, 737–756 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Damlamian, A.: Some results on the multi-phase Stefan problem. Commun. Partial Differ. Equ. 2, 1017–1044 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Damlamian, A., Kenmochi, N.: Evolution equations generated by subdifferentials in the dual space of \((H^1(\Omega ))\). Discrete Contin. Dyn. Syst. 5, 269–278 (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fukao, T.: Convergence of Cahn–Hilliard systems to the Stefan problem with dynamic boundary conditions. Asymptot. Anal. 99, 1–21 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fukao, T.: Cahn–Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions. In: Bociu, L., Désidéri, J.-A., Habbal, A. (eds.) System Modeling and Optimization, pp. 282–291. Springer, Basel (2016)Google Scholar
  13. 13.
    Fukao, T., Kurima, S., Yokota, T.: Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems on unbounded domains via Cauchy’s criterion. Math. Methods Appl. Sci. 1–12 (2018).
  14. 14.
    Fukao, T., Motoda, T.: Abstract approach of degenerate parabolic equations with dynamic boundary conditions. Adv. Math. Sci. Appl. 27, 29–44 (2018)Google Scholar
  15. 15.
    Kenmochi, N.: Neumann problems for a class of nonlinear degenerate parabolic equations. Differ. Integral Equ. 2, 253–273 (1990)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kenmochi, N., Niezgódka, M., Pawłow, I.: Subdifferential operator approach to the Cahn–Hilliard equation with constraint. J. Differ. Equ. 117, 320–354 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kubo, M.: The Cahn–Hilliard equation with time-dependent constraint. Nonlinear Anal. 75, 5672–5685 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kubo, M., Lu, Q.: Nonlinear degenerate parabolic equations with Neumann boundary condition. J. Math. Anal. Appl. 307, 232–244 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kurima, S., Yokota, T.: Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates. J. Differ. Equ. 263, 2024–2050 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kurima, S., Yokota, T.: A direct approach to quasilinear parabolic equations on unbounded domains by Brézis’s theory for subdifferential operators. Adv. Math. Sci. Appl. 26, 221–242 (2017)Google Scholar
  21. 21.
    Miranville, A.: The Cahn–Hilliard equation and some of its variants. AIMS Math. 2, 479–544 (2017)CrossRefGoogle Scholar
  22. 22.
    Simon, J.: Compact sets in the spaces \(L^p(0, T;B)\). Ann. Mat. Pura. Appl. 4(146), 65–96 (1987)zbMATHGoogle Scholar

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

Personalised recommendations