• DOI:10.1002/mma.4760
• Corpus ID: 53539862

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

@article{Fukao2017NonlinearDE,
  title={Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion},
  author={Takeshi Fukao and Shunsuke Kurima and Tomomi Yokota},
  journal={Mathematical Methods in the Applied Sciences},
  year={2017},
  volume={41},
  pages={2590 - 2601}
}

• T. Fukao, Shunsuke Kurima, T. Yokota
• Published 27 November 2015
• Mathematics
• Mathematical Methods in the Applied Sciences

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem for the nonlinear diffusion equation in an unbounded domain Ω⊂RN ( N∈N ), written as ∂u∂t+(−Δ+1)β(u)=ginΩ×(0,T), which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on R , and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem ε with approximate parameter ε>0: ∂uε∂t+(−Δ+1)(ε(−Δ+1)uε+β(uε)+πε(uε))=ginΩ×(0,T), which is called the Cahn‐Hilliard system, even if Ω⊂RN ( N∈N ) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.