Singularly perturbed 1D Cahn–Hilliard equation revisited

Abstract

We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors \({\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}\) being the perturbation parameter, such that the map \({\epsilon \mapsto {\mathcal M}_\epsilon}\) is Hölder continuous. Besides, the continuity at \({\epsilon=0}\) is obtained with respect to a metric independent of \({\epsilon.}\) Continuity properties of global attractors and inertial manifolds are also examined.