Attractors for semigroups with multi-dimensional time and PDEs in unbounded domains

TL;DR

This paper develops a new attractors theory for semigroups with multi-dimensional time in Euclidean spaces and applies it to PDEs in unbounded domains, including exterior domains, without domain shape restrictions.

Contribution

It introduces a general attractors framework for multidimensional time semigroups applicable to PDEs in arbitrary unbounded domains, removing previous geometric constraints.

Findings

Established attractors theory for multidimensional time semigroups.

Applied the theory to elliptic PDEs in unbounded domains.

Extended applicability to exterior and non-cylindrical domains.

Abstract

We develop the attractors theory for the semigroups with multidimensional time belonging to some closed cone in an Euclidean space and apply the obtained general results to partial differential equations (PDEs) in unbounded domains. The main attention is payed to elliptic boundary problems in general unbounded domains. In contrast to the previous works in this direction our theory does not require the underlying domain to be cylindrical or cone-like or to be shift semi-invariant with respect to some direction. In particular, the theory is applicable to the exterior domains.