The authors consider the generalised Cahn-Hilliard equation
which gives rise to four cases: (a) , , (b) , , (c) , (d) , . Since cases (a), (b) are considered as settled, the authors concentrate on (c), (d). The aim is to show that the global attractor , induced by the semiflow associated with (1), is upper semicontinuous with respect to the parameters , . 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 -periodic functions such as are introduced, in terms of which the spaces underlying (1) are defined such as in particular . 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 , is obtained, thus providing a semiflow for (1). Theorem 3.4, based on Theorem 2.1 and some topological considerations, states that gives rise to a global attractor ; moreover if , a global attractor exists. After a series of propositions, further results on the existence and structure of such as boundedness (Theorem 3.6) are obtained.
One of the main result is expressed by Theorem 3.7. With the global attractor of (1) for , , it states that the family is upper semicontinuous at with respect to the topology of the fractional power space , defined in terms of the Laplacian .
Further results of a similar type are expressed by Theorems 3.8 and 3.9. Some technical steps are relegated to an appendix.