Totally dissipative dynamical processes and their uniform global attractors

TL;DR

This paper investigates the existence of uniform global attractors for totally dissipative dynamical processes on metric spaces, without requiring continuity, and provides characterizations under mild additional conditions.

Contribution

It establishes the existence of uniform global attractors for totally dissipative processes without continuity assumptions and offers characterizations when mild conditions are met.

Findings

Existence of uniform global attractors for totally dissipative processes.

Attractors are uniform with respect to parameters and initial times.

Characterization of attractors under mild continuity-like hypotheses.

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 $\tau\in\R$. 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.