On the continuity of global attractors

Abstract

Let $\Lambda$ be a complete metric space, and let $\{S_{\lambda }(\cdot ):\lambda \in \Lambda \}$ be a parametrised family of semigroups with global attractors ${\mathcal{A}}_{\lambda }$. We assume that there exists a fixed bounded set $D$ such that ${\mathcal{A}}_{\lambda }\subset D$ for every $\lambda \in \Lambda$. By viewing the attractors as the limit as $t\to \infty$ of the sets $S_{\lambda }(t)D$, we give simple proofs of the equivalence of `equi-attraction' to continuity (when this convergence is uniform in $\lambda$) and show that the attractors ${\mathcal{A}}_{\lambda }$ are continuous in $\lambda$ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise).