Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems

TL;DR

This paper extends the concept of principal eigenvalues to space-time periodic cooperative systems, addressing challenges in unbounded domains and exploring properties and optimization of these generalized eigenvalues.

Contribution

It introduces a new family of generalized principal eigenvalues for cooperative systems, establishing existence, uniqueness, and optimization properties in the vector setting.

Findings

Existence and uniqueness of principal eigenpairs in space-time periodic systems.

Development of a one-parameter family of eigenvalues related to eigenfunctions.

An optimization property for mutation operators in bistochastic matrices.

Abstract

This paper is concerned with generalizations of the notion of principal eigenvalue in the context of space-time periodic cooperative systems. When the spatial domain is the whole space, the Krein-Rutman theorem cannot be applied and this leads to more sophisticated constructions and to the notion of generalized principal eigenvalues. These are not unique in general and we focus on a one-parameter family corresponding to principal eigenfunctions that are space-time periodic multiplicative perturbations of exponentials of the space variable. Besides existence and uniqueness properties of such principal eigenpairs, we also prove various dependence and optimization results illustrating how known results in the scalar setting can, or cannot, be extended to the vector setting. We especially prove an optimization property on minimizers and maximizers among mutation operators valued in the set of bistochastic matrices that is, to the best of our knowledge, new.