### 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 ⊂ ℝ^{N} (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 ⊂ ℝ^{N} (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 language | English |
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Pages (from-to) | 2590-2601 |

Number of pages | 12 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 41 |

Issue number | 7 |

DOIs | |

Publication status | Published - 15 May 2018 |

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### Keywords

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