Global attractors for singular perturbations of the Cahn-Hilliard equations
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 \(\varepsilon=0\), \(\delta>0\), it states that the family \(\{A_{\varepsilon\delta}, 0\leq\varepsilon\leq \varepsilon_1\}\) is upper semicontinuous at \(\varepsilon=0\) with respect to the topology of the fractional power space \(X_{2-\eta}\), defined in terms of the Laplacian \(-\Delta\). Further results of a similar type are expressed by Theorems 3.8 and 3.9. Some technical steps are relegated to an appendix.
scientific article; zbMATH DE number 2136641
scientific article