# 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} }

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…

## 4 Citations

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## References

SHOWING 1-10 OF 71 REFERENCES

Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates

- Mathematics
- 2017

Convergence in C([0,T];L 2 (Ω)) of weak solutions to perturbed doubly degenerate parabolic equations

- Mathematics
- 2016

Well-posedness of singular diffusion equations in porous media with homogeneous Neumann boundary conditions

- Mathematics
- 2010

Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations

- Mathematics
- 2015

CHAPTER 4 - Monotonicity and Compactness Methods for Nonlinear Variational Inequalities

- Mathematics
- 2007

BOUNDED WEAK SOLUTIONS OF AN ELLIPTIC-PARABOLIC NEUMANN PROBLEM

- Mathematics
- 1987

ABSTRACT. In this paper we establish existence and uniqueness for boundedweak solutions of an elliptic-parabolic Neumann problem. We also describethe asymptotic behavior as t —> oo. 1. Introduction.…

Obstructions to Existence in Fast-Diffusion Equations

- Mathematics
- 2002

Abstract The study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory. Thus, in the sub-field of very fast diffusion it is known that…

A direct approach to quasilinear parabolic equations on unbounded domains by Br\'ezis's theory for subdifferential operators

- Mathematics
- 2017

This paper is concerned with existence and uniqueness of solutions to two kinds of quasilinear parabolic equations. One is described as the form which includes the porous media and fast diffusion…

LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS

- Mathematics
- 2001

The purpose of this paper is to study the limit in L 1(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form ut − Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary…