WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION

doi:10.1007/s10473-025-0411-4

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1438-1470.doi: 10.1007/s10473-025-0411-4

WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION

Siyan GUO¹, Jiangbo HAN^1,2, Runzhang XU^1,*

1. College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China;
2. School of Mathematical Sciences, Inner Mongolia University, Inner Mongolia 010030, China

收稿日期:2024-02-19 修回日期:2024-06-14 出版日期:2025-10-10 发布日期:2025-10-10

WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION

Siyan GUO¹, Jiangbo HAN^1,2, Runzhang XU^1,*

1. College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China;
2. School of Mathematical Sciences, Inner Mongolia University, Inner Mongolia 010030, China

Received:2024-02-19 Revised:2024-06-14 Online:2025-10-10 Published:2025-10-10
Contact: ^*Runzhang XU, E-mail: xurunzh@163.com; xurunzh@hrbeu.edu.cn
About author:Siyan GUO, E-mail: guosiyansiyan@163.com; guosiyan@hrbeu.edu.cn; Jiangbo HAN, E-mail: han1825141364@163.com; 2014112111@hrbeu.edu.cn
Supported by:
NSFC (12271122), and the Fundamental Research Funds for the Central Universities. Han's research was supported by the Fun-damental Research Funds for the Central Universities (3072023GIP2401).

摘要/Abstract

摘要： In this paper, we consider the initial boundary value problem for the 2-D hyperbolic viscous Cahn-Hilliard equation. Firstly, we prove the existence and uniqueness of the local solution by the Galerkin method and contraction mapping principle. Then, using the potential well theory, we study the global well-posedness of the solution with initial data at different levels of initial energy, i.e., subcritical initial energy, critical initial energy and arbitrary positive initial energy. For subcritical initial energy, we prove the global existence, asymptotic behavior and finite time blowup of the solution. Moreover, we extend these results to the critical initial energy using the scaling technique. For arbitrary positive initial energy, including the sup-critical initial energy, we obtain the sufficient conditions for finite time blow-up of the solution. As a further study for estimating the blowup time, we give a unified expression of the lower bound of blowup time for all three initial energy levels and estimate the upper bound of blowup time for subcritical and critical initial energy.

关键词: 2-D hyperbolic viscous Cahn-Hilliard equation, global existence, finite time blow up, exponential decay

Abstract: In this paper, we consider the initial boundary value problem for the 2-D hyperbolic viscous Cahn-Hilliard equation. Firstly, we prove the existence and uniqueness of the local solution by the Galerkin method and contraction mapping principle. Then, using the potential well theory, we study the global well-posedness of the solution with initial data at different levels of initial energy, i.e., subcritical initial energy, critical initial energy and arbitrary positive initial energy. For subcritical initial energy, we prove the global existence, asymptotic behavior and finite time blowup of the solution. Moreover, we extend these results to the critical initial energy using the scaling technique. For arbitrary positive initial energy, including the sup-critical initial energy, we obtain the sufficient conditions for finite time blow-up of the solution. As a further study for estimating the blowup time, we give a unified expression of the lower bound of blowup time for all three initial energy levels and estimate the upper bound of blowup time for subcritical and critical initial energy.

Key words: 2-D hyperbolic viscous Cahn-Hilliard equation, global existence, finite time blow up, exponential decay

Siyan GUO, Jiangbo HAN, Runzhang XU. WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION[J]. 数学物理学报(英文版), 2025, 45(4): 1438-1470.

Siyan GUO, Jiangbo HAN, Runzhang XU. WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1438-1470.

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WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION

WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION

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