### Upper semicontinuity of global attractors for the perturbed viscous Cahn-Hilliard equations

DOI: http://dx.doi.org/10.12775/TMNA.2008.051

#### Abstract

It is known that the semigroup generated by the initial-boundary

value problem for the perturbed viscous Cahn-Hilliard equation with

$\varepsilon> 0$ as a parameter

admits a global attractor $\mathcal{A}_{\varepsilon}$ in the phase

space $X^{{1}/{2}} =(H^2(\Omega)\cap H^{1}_{0}(\Omega))\times L^2(\Omega)$,

$\Omega\subset \mathbb{R}^n$, $n\leq 3$ (see [M. B. Kania,

< i> Global attractor for the perturbed viscous Cahn-Hilliard equation< /i> , Colloq.

Math. < b> 109< /b> (2007), 217-229]). In this paper

we show that the family $\{\mathcal{A}_{\varepsilon}\}_{\varepsilon\in[0,1]}$

is upper semicontinuous at $0$, which means that the Hausdorff semidistance

$$

d_{X^{{1}/{2}}}(\mathcal{A}_{\varepsilon},\mathcal{A}_0)\equiv

\sup_{\psi\in

\mathcal{A}_{\varepsilon}}\inf_{\phi\in\mathcal{A}_{0}}\|

\psi-\phi\|_{X^{{1}/{2}}},

$$

tends to 0 as $\varepsilon\to 0^{+}$.

value problem for the perturbed viscous Cahn-Hilliard equation with

$\varepsilon> 0$ as a parameter

admits a global attractor $\mathcal{A}_{\varepsilon}$ in the phase

space $X^{{1}/{2}} =(H^2(\Omega)\cap H^{1}_{0}(\Omega))\times L^2(\Omega)$,

$\Omega\subset \mathbb{R}^n$, $n\leq 3$ (see [M. B. Kania,

< i> Global attractor for the perturbed viscous Cahn-Hilliard equation< /i> , Colloq.

Math. < b> 109< /b> (2007), 217-229]). In this paper

we show that the family $\{\mathcal{A}_{\varepsilon}\}_{\varepsilon\in[0,1]}$

is upper semicontinuous at $0$, which means that the Hausdorff semidistance

$$

d_{X^{{1}/{2}}}(\mathcal{A}_{\varepsilon},\mathcal{A}_0)\equiv

\sup_{\psi\in

\mathcal{A}_{\varepsilon}}\inf_{\phi\in\mathcal{A}_{0}}\|

\psi-\phi\|_{X^{{1}/{2}}},

$$

tends to 0 as $\varepsilon\to 0^{+}$.

#### Keywords

Perturbed viscous Cahn-Hilliard equation; global attractor; upper semicontinuity

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