Totally dissipative dynamical processes and their uniform global attractors

Abstract

We discuss the existence of the global attractor for a family of processes $U_{\sigma }(t,\tau )$ acting on a metric space $X$ and depending on a symbol $\sigma$ belonging to some other metric space $\Sigma$. Such an attractor is uniform with respect to $\sigma \in \Sigma$, as well as with respect to the choice of the initial time $\tau \in \mathbb{R}$. The existence of the attractor is established for totally dissipative processes without any continuity assumption. When the process satisfies some additional (but rather mild) continuity-like hypotheses, a characterization of the attractor is given.