Evolution equations with eventually positive solutions

TL;DR

This paper investigates linear evolution equations where solutions starting positive eventually become and remain positive, aiming to develop a general theory for this phenomenon beyond specific spectral conditions.

Contribution

It introduces a general framework to understand eventual positivity in linear evolution equations without relying on restrictive spectral assumptions.

Findings

Identification of conditions leading to eventual positivity

Extension of existing theories to broader classes of equations

Insights into long-term behavior of solutions

Abstract

We discuss linear autonomous evolution equations on function spaces which have the property that a positive initial value leads to a solution which initially changes sign, but then becomes - and stays - positive again for sufficiently large times. This eventual positivity phenomenon has recently been discovered for various classes of differential equations, but so far a general theory to explain this type of behaviour exists only under additional spectral assumptions.