Convergence to equilibrium for the Cahn–Hilliard equation with dynamic boundary conditions

• Published in: Journal of Differential Equations Vol. 204; no. 2; pp. 511 - 531
• Main Authors: Wu, Hao, Zheng, Songmu
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 20.09.2004
• Subjects: Cahn–Hilliard equation; Dynamic boundary condition; Simon–Lojasiewicz inequality; Dynamic boundary condition; Cahn–Hilliard equation; Simon–Lojasiewicz inequality
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2004.05.004

This paper is concerned with the asymptotic behavior of solution to the Cahn–Hilliard equation (0.1) ∂u ∂t = Δμ, μ=− Δu−u+u 3, (x,t)∈Ω×R + subject to the following dynamic boundary conditions: (0.2) σ s Δ ||u−∂ νu+h s −g s u= 1 Γ s u t, t>0, x∈Γ, ∂ νμ=0, t>0, x∈Γ and the initial condition (0.3) u| t=0=u 0(x), x∈Ω, where Ω is a bounded domain in R n (n⩽3) with smooth boundary Γ , and Γ s>0, σ s>0, g s>0, h s are given constants; Δ || is the tangential Laplacian operator, and ν is the outward normal direction to the boundary. This problem has been considered in the recent paper by Racke and Zheng (Adv. Differential Equations 8 (1) (2003) 83) where the global existence and uniqueness were proved. In a very recent manuscript by Prüss, Racke and Zheng (Konstanzer Schrift. Math. Inform. 189 (2003)) the results on existence of global attractor and maximal regularity of solution have been obtained. In this paper, convergence of solution of this problem to an equilibrium, as time goes to infinity, is proved.