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Global attractors for singular perturbations of the Cahn-Hilliard equations. (English) Zbl 1063.35041

The authors consider the generalised Cahn-Hilliard equation

εu tt +Δ(Δu-u 3 +u-δu t )=0,x[0,π],u(0,t)=u(π,t)(1)

which gives rise to four cases: (a) ε=0, δ=0, (b) ε=0, δ>0, (c) ε>0, δ=0 (d) ε>0, δ>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 εδ , induced by the semiflow S εδ R associated with (1), is upper semicontinuous with respect to the parameters ε, δ. 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 π-periodic functions such as H m =H m (0,π)H 0 1 (0,π) are introduced, in terms of which the spaces underlying (1) are defined such as in particular X=H 1 ×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 ε(0,1], δ(0,1] is obtained, thus providing a semiflow S εδ for (1). Theorem 3.4, based on Theorem 2.1 and some topological considerations, states that S εδ gives rise to a global attractor A εδ (ε(0,1],δ(0,1]); moreover if ε1/3, δ=0 a global attractor A ε0 exists. After a series of propositions, further results on the existence and structure of A εδ such as boundedness (Theorem 3.6) are obtained.

One of the main result is expressed by Theorem 3.7. With A 0δ the global attractor of (1) for ε=0, δ>0, it states that the family {A εδ ,0εε 1 } is upper semicontinuous at ε=0 with respect to the topology of the fractional power space X 2-η , defined in terms of the Laplacian -Δ.

Further results of a similar type are expressed by Theorems 3.8 and 3.9. Some technical steps are relegated to an appendix.

MSC:
35B41Attractors (PDE)
35B40Asymptotic behavior of solutions of PDE
35L70Nonlinear second-order hyperbolic equations
37L30Attractors and their dimensions, Lyapunov exponents
35Q53KdV-like (Korteweg-de Vries) equations
35B25Singular perturbations (PDE)