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

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, Global attractor for the perturbed viscous Cahn–Hilliard equation, Colloq. Math. 109 (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 $\varepsilon\to 0^{+}$.