Autonomous and non-autonomous unbounded attractors in evolutionary problems

TL;DR

This paper extends the theory of unbounded attractors to slowly non-dissipative evolutionary problems, providing abstract results, properties, and criteria for their structure, with applications to autonomous equations.

Contribution

It develops the unbounded attractor theory for non-autonomous and autonomous problems, including criteria for attractor structure using inertial manifolds.

Findings

Established existence of unbounded attractors in non-dissipative settings

Analyzed properties of unbounded ω-limit sets

Provided criteria for attractor structure via Lipschitz graphs

Abstract

If the semigroup is slowly non-dissipative, i.e., its solutions can diverge to infinity as time tends to infinity, one still can study its dynamics via the approach by the unbounded attractors - the counterpart of the classical notion of global attractors. We continue the development of this theory started by Chepyzhov and Goritskii [CG92]. We provide the abstract results on the unbouded attractor existence, and we study the properties of these attractors, as well as of unbounded $\omega$-limit sets in slowly non-dissipative setting. We also develop the pullback non-autonomous counterpart of the unbounded attractor theory. The abstract theory that we develop is illustrated by the analysis of the autonomous problem governed by the equation $u_t = Au + f(u)$. In particular, using the inertial manifold approach, we provide the criteria under which the unbounded attractor coincides with the graph of the Lipschitz function, or becomes close to the graph of the Lipschitz function for large argument.