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 ${\mathscr A}_\lambda$. We assume that there exists a fixed bounded set $D$ such that ${\mathscr A}_\lambda\subset D$ 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 ${\mathscr A}_\lambda$ are continuous in $\lambda$ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise).