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

Takeshi Fukao, Shunsuke Kurima, Tomomi Yokota

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4 Citations (Scopus)

Abstract

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 (Formula presented.) 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: (Formula presented.) 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.

Original languageEnglish
Pages (from-to)2590-2601
Number of pages12
JournalMathematical Methods in the Applied Sciences
Volume41
Issue number7
DOIs
Publication statusPublished - 15 May 2018

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Keywords

  • Cahn-Hilliard systems
  • Stefan problems
  • fast diffusion equations
  • nonlinear diffusion equations
  • porous media equations
  • subdifferential operators

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