有界域上四阶 Cahn-Hilliard 方程在二重特征值处的定态分歧

Abstract

Abstract:

Employing the normalized Lyapunov-Schmidt reduction method and the spectral decomposition theorem for linear completely continuous fields to investigate the steady-state bifurcation of the Cahn-Hilliard equation. On a square bounded domain, we prove that the fourth-order Cahn-Hilliard equation undergoes steady-state bifurcation at the first eigenvalue (double) under homogeneous Dirichlet boundary conditions and homogeneous Robin boundary conditions. In such cases, the Cahn-Hilliard equation admits nontrivial solutions. Furthermore, the complete criteria for supercritical and subcritical bifurcations under both boundary conditions, explicit expressions for the bifurcation solutions, regularity of bifurcation solutions and the bifurcation solutions' diagrams were obtained.

Key words: Cahn-Hilliard equation, Dirichlet boundary, Robin boundary, linear completely continuous fields, Lyapunov-Schmidt reduction

CLC Number:

O175.29

Ruilin Liao, Fang Guan, Zhigang Pan. Steady-State Bifurcation From Double Eigenvalue for the Fourth-Order Cahn-Hilliard Equation on a Bounded Domain[J].Acta mathematica scientia,Series A, 2026, 46(5): 1751-1768.

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Figures/Tables 16

References 34

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Steady-State Bifurcation From Double Eigenvalue for the Fourth-Order Cahn-Hilliard Equation on a Bounded Domain

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