Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn–Hilliard systems
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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.
KeywordsCahn–Hilliard system Degenerate parabolic equation Robin boundary condition Growth condition
Mathematics Subject Classification35K61 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 (http://www.edanzediting.com/ac) for editing a draft of this manuscript.
- 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
- 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.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). https://doi.org/10.1002/mma.4760
- 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
- 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