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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 $$\varepsilon u_{tt} + \Delta(\Delta u-u^3 + u - \delta u_t) = 0, \quad x\in [0,\pi],\quad u(0,t) = u(\pi,t)\tag1$$ 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 [{\it S. Zheng} and {\it 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.

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)
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References:
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