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).