Nonautonomous systems of evolution inclusions

TL;DR

This paper establishes the existence of solutions for complex coupled nonautonomous evolution inclusions, including Schrödinger-Debye and Maxwell-parabolic systems, using advanced semigroup and set-valued analysis techniques.

Contribution

It extends existing methods to nonautonomous systems, introducing new existence results for measurable selections and handling set-valued coupling terms with Hausdorff continuity.

Findings

Proves existence of global solutions for coupled evolution inclusions.

Includes nonautonomous Schrödinger-Debye and Maxwell-parabolic systems.

Extends semigroup approach to nonautonomous and set-valued contexts.

Abstract

We prove the existence of global solutions for some coupled systems of partially nonautonomous evolution inclusions comprised of a Cauchy problem with a compact resolvent semigroup generator and an evolution equation governed by a subdifferential of a real potential. Our system in particular includes nonautonomous generalized Schr\"odinger-Debye systems of inclusions with variable exponents, but extends to hyperbolic-parabolic systems of inclusions in particular to Maxwell-parabolic systems of inclusions. Methodologically, we extend an approach of Vrabie et al. to the nonautonomous case and make use of standard semigroup tools to accomodate non-parabolic behaviour of solutions paired with a new existence result for measurable selections. The combination of the latter two requires the set-valued coupling terms to be Hausdorff-continuous, to take bounded, convex and closed values, and to satisfy weak continuity with respect to one variable.