# Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator

## TL;DR

This paper studies how the principal eigenvalue of a time-periodic parabolic operator changes with frequency, establishing monotonicity and asymptotic behavior, and confirming a conjecture from prior mathematical biology research.

## Contribution

It proves the monotonicity and asymptotic properties of the principal eigenvalue with respect to frequency, confirming a conjecture in the field.

## Key findings

- Principal eigenvalue's monotonicity with frequency established.
- Asymptotic behavior of eigenvalue analyzed.
- Conjecture by Hutson et al. confirmed.

## Abstract

We investigate the effect of frequency on the principal eigenvalue of a time-periodic parabolic operator with Dirichlet, Robin or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the principal eigenvalue with respect to the frequency parameter are established. Our results prove a conjecture raised by Hutson, Michaikow and Pol\'{a}\v{c}ik [2001 J. Math. Biol.].