A criterion for the uniform eventual positivity of operator semigroups

TL;DR

This paper establishes a new criterion to determine when the operators in a $C_0$-semigroup become positive after some time, even if the entire semigroup is not initially positive, with applications to PDEs.

Contribution

It introduces a sufficient condition for the eventual positivity of operators in a semigroup on Banach lattices, expanding understanding of long-term positivity behavior.

Findings

Provides a criterion for uniform eventual positivity of operator semigroups.

Applies the criterion to analyze PDE solution behaviors.

Enhances the theoretical framework for positivity in semigroup theory.

Abstract

Consider a $C_0$-semigroup $(e^{tA})_{t \ge 0}$ on a function space or, more generally, on a Banach lattice $E$. We prove a sufficient criterion for the operators $e^{tA}$ to be positive for all sufficiently large times $t$, while the semigroup itself might not be positive. This complements recently established criteria for the individual orbits of the semigroup to become eventually positive for all positive initial values. We apply our main result to study the qualitative behaviour of the solutions to various partial differential equations.