Convergence to Equilibrium for the Cahn-Hilliard Equation with Wentzell Boundary Condition
Convergence to Equilibrium for the Cahn-Hilliard Equation with Wentzell Boundary Condition
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DOI：
10.1016/j.jde.2004.05.004
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发表时间：
2004-09
期刊：
影响因子：
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通讯作者：
Hao-qing Wu
中科院分区：
文献类型：
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作者：
Hao-qing Wu
本文研究了Cahn-Hilliard方程[公式：见正文]在下列动态边界条件下解的渐近性态：[公式：见正文]和初始条件[公式：见正文]其中Ω是Rn（n <$3）中具有光滑边界的有界区域，且Γ s> 0，σ s> 0，gs> 0，hs为给定常数; Δ||是切向拉普拉斯算子，ν是边界的向外法线方向。Racke和Zheng在最近的论文（Adv.Differential Equations 8（1）（2003）83）中已经考虑了这个问题，其中证明了全局存在性和唯一性。在Prüss，Racke和Zheng（Konstanzer Schrift.数学。信息。189（2003））得到了整体吸引子的存在性和解的极大正则性的结果.本文证明了当时间趋于无穷大时，该问题的解收敛于一个平衡点。
This paper is concerned with the asymptotic behavior of solution to the Cahn–Hilliard equation [Formula: see text] subject to the following dynamic boundary conditions: [Formula: see text] and the initial condition [Formula: see text] where Ω is a bounded domain in Rn(n⩽3) with smooth boundary Γ , and Γs>0, σs>0, gs>0, hsare 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.