On the stability and transition of the Cahn-Hilliard/Allen-Cahn system

Quan Wang; Dongming Yan

doi:10.3934/dcdsb.2020024

Article Contents

2020, Volume 25, Issue 7: 2607-2620. Doi: 10.3934/dcdsb.2020024

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$ p(x) $

-Laplacian Steklov eigenvalue problem

On the stability and transition of the Cahn-Hilliard/Allen-Cahn system

Quan Wang^a, and
Dongming Yan^b, ,

a.
Department of Mathematics, Sichuan University, Chengdu 610065, China
b.
School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou 310018, China

^* Corresponding author: Dongming Yan
^* Corresponding author: Dongming Yan

Received: December 31, 2018

Early access: April 2020

Published: July 2020

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Abstract

In this paper, the main objective is to study the stability and transition of the Cahn-Hilliard/Allen-Cahn system. By using the dynamic transition theory, combining with the spectral theorem for general linear completely continuous fields, we prove that the system undergoes a continuous transition and bifurcates from a trivial solution to an attractor as the control parameter crosses a certain critical value. In addition, for some special cases, i.e., the domain is $ n $-dimensional box $ (n = 1,2,3) $, we not only obtain the stability of the singular points of the attractors, the topological structure of the attractors is also illustrated.

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Mathematics Subject Classification: Primary: 35G60; Secondary: 35Q56.

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Figure 1. The topological structure of attractor for $ n = 1 $

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Figure 2. The topological structure of attractor for $ n = 2 $

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Figure 3. The topological structure of attractor for $ n = 3 $

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