Global attractors for singular perturbations of the Cahn-Hilliard equations
type: Item
instance of: scholarly article
title: Global attractors for singular perturbations of the Cahn-Hilliard equations
zbMATH Open document ID: 1063.35041
author: Songmu Zheng
author: Albert Milani
published in: Journal of Differential Equations
publication date: 2005-02-22T00:00:00Z
review text: The authors consider the generalised Cahn-Hilliard equation \[ \varepsilon u_{tt} + \Delta(\Delta u-u^3 + u - \delta u_t) = 0, \quad x\in [0,\pi],\quad u(0,t) = u(\pi,t)\tag{1} \] which gives rise to four cases: (a) \(\varepsilon=0\), \(\delta=0\), (b) \(\varepsilon=0\), \(\delta>0\), (c) \(\varepsilon>0\), \(\delta=0\) (d) \(\varepsilon>0\), \(\delta >0\). Since cases (a), (b) are considered as settled, the authors concentrate on (c), (d). The aim is to show that the global attractor \(A_{\varepsilon\delta}\), induced by the semiflow \(S_{\varepsilon\delta}R\) associated with (1), is upper semicontinuous with respect to the parameters \(\varepsilon\), \(\delta\). The paper is based on an earlier one [\textit{S. Zheng} and \textit{A. Milani}, Nonlinear Anal., Theory Methods Appl. 57A, No. 5-6, 843--877 (2004; Zbl 1055.35028)] in which a number of results on (1) are obtained. In order to handle (1), Sobolev spaces of \(\pi\)-periodic functions such as \(H^m=H^m(0,\pi)\cap H^1_0(0,\pi)\) are introduced, in terms of which the spaces underlying (1) are defined such as in particular \(X =H^1\times H^{-1}\). After a definition of weak solutions of (1) and a number of propositions, Theorem 2.1, asserting global existence and uniqueness of weak solutions of (1) for \(\varepsilon\in (0,1]\), \(\delta\in (0,1]\) is obtained, thus providing a semiflow \(S_{\varepsilon\delta}\) for (1). Theorem 3.4, based on Theorem 2.1 and some topological considerations, states that \(S_{\varepsilon\delta}\) gives rise to a global attractor \(A_{\varepsilon\delta}\) \((\varepsilon\in (0,1],\delta\in (0,1])\); moreover if \(\varepsilon\leq 1/3\), \(\delta=0\) a global attractor \(A_{\varepsilon 0}\) exists. After a series of propositions, further results on the existence and structure of \(A_{\varepsilon\delta}\) such as boundedness (Theorem 3.6) are obtained. One of the main result is expressed by Theorem 3.7. With \(A_{0\delta}\) the global attractor of (1) for \(\varepsilon=0\), \(\delta>0\…
reviewed by: Bruno Scarpellini
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 2136641
zbMATH Keywords: upper semicontinuity
zbMATH Keywords: global existence and uniqueness
zbMATH Keywords: weak solutions
MaRDI profile type: Publication
full work available at URL: j.jde.2004.08.026
OpenAlex ID: W2066966173
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DOI: 10.1016/J.JDE.2004.08.026
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description: scientific article; zbMATH DE number 2136641