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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

$\epsilon {u}_{tt}+{\Delta }\left({\Delta }u-{u}^{3}+u-\delta {u}_{t}\right)=0,\phantom{\rule{1.em}{0ex}}x\in \left[0,\pi \right],\phantom{\rule{1.em}{0ex}}u\left(0,t\right)=u\left(\pi ,t\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

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 ${H}^{m}={H}^{m}\left(0,\pi \right)\cap {H}_{0}^{1}\left(0,\pi \right)$ 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 $\epsilon \in \left(0,1\right]$, $\delta \in \left(0,1\right]$ 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 ${A}_{\epsilon \delta }$ $\left(\epsilon \in \left(0,1\right],\delta \in \left(0,1\right]\right)$; 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 $\left\{{A}_{\epsilon \delta },0\le \epsilon \le {\epsilon }_{1}\right\}$ 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.

##### MSC:
 35B41 Attractors (PDE) 35B40 Asymptotic behavior of solutions of PDE 35L70 Nonlinear second-order hyperbolic equations 37L30 Attractors and their dimensions, Lyapunov exponents 35Q53 KdV-like (Korteweg-de Vries) equations 35B25 Singular perturbations (PDE)