*(English)*Zbl 1063.35041

The authors consider the generalised Cahn-Hilliard equation

which gives rise to four cases: (a) $\epsilon =0$, $\delta =0$, (b) $\epsilon =0$, $\delta >0$, (c) $\epsilon >0$, $\delta =0$ (d) $\epsilon >0$, $\delta >0$. Since cases (a), (b) are considered as settled, the authors concentrate on (c), (d). The aim is to show that the global attractor ${A}_{\epsilon \delta}$, induced by the semiflow ${S}_{\epsilon \delta}R$ associated with (1), is upper semicontinuous with respect to the parameters $\epsilon $, $\delta $. The paper is based on an earlier one [*S. Zheng* and *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 ${H}^{m}={H}^{m}(0,\pi )\cap {H}_{0}^{1}(0,\pi )$ 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 $\epsilon \in (0,1]$, $\delta \in (0,1]$ is obtained, thus providing a semiflow ${S}_{\epsilon \delta}$ for (1). Theorem 3.4, based on Theorem 2.1 and some topological considerations, states that ${S}_{\epsilon \delta}$ gives rise to a global attractor ${A}_{\epsilon \delta}$ $(\epsilon \in (0,1],\delta \in (0,1\left]\right)$; moreover if $\epsilon \le 1/3$, $\delta =0$ a global attractor ${A}_{\epsilon 0}$ exists. After a series of propositions, further results on the existence and structure of ${A}_{\epsilon \delta}$ 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 $\epsilon =0$, $\delta >0$, it states that the family $\{{A}_{\epsilon \delta},0\le \epsilon \le {\epsilon}_{1}\}$ is upper semicontinuous at $\epsilon =0$ 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.