Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator II: Small diffusion

TL;DR

This paper analyzes how small diffusion influences the principal eigenvalues of linear time-periodic parabolic operators in one dimension, revealing dependence on periodic solutions of an induced ODE.

Contribution

It provides new asymptotic results for eigenvalues in small diffusion limits, including both non-degenerate and degenerate environments, highlighting the role of associated periodic ODE solutions.

Findings

Asymptotic behavior of eigenvalues as diffusion tends to zero

Dependence of eigenvalue asymptotics on periodic ODE solutions

Extension to both non-degenerate and degenerate environments

Abstract

We investigate the effect of small diffusion on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions in one dimensional space. The asymptotic behaviors of the principal eigenvalues, as the diffusion coefficients tend to zero, are established for non-degenerate and degenerate spatial-temporally varying environments. A new finding is the dependence of these asymptotic behaviors on the periodic solutions of a specific ordinary differential equation induced by the drift. The proofs are based upon delicate constructions of super/sub-solutions and the applications of comparison principles.