Principal eigenvalue and maximum principle for cooperative periodic-parabolic systems

TL;DR

This paper characterizes the positivity and maximum principle for cooperative periodic-parabolic systems by linking supersolutions, principal eigenvalues, and resolvent operators, providing a comprehensive understanding of the system's positivity properties.

Contribution

It introduces a classification of supersolutions and establishes a novel scalar characterization of the maximum principle for cooperative systems using principal eigenvalues.

Findings

Positivity of the resolvent operator is characterized by the principal eigenvalue.

Supersolution classification helps determine maximum principles.

The results apply under arbitrary boundary conditions of mixed type.

Abstract

This paper classifies the set of supersolutions of a general class of periodic-parabolic problems in the presence of a positive supersolution. From this result we characterize the positivity of the underlying resolvent operator through the positivity of the associated principal eigenvalue and the existence of a positive strict supersolution. Lastly, this (scalar) characterization is used to characterize the strong maximum principle for a class of periodic-parabolic systems of cooperative type under arbitrary boundary conditions of mixed type.