Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator I: Large advection

TL;DR

This paper analyzes how large advection influences the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions, revealing various asymptotic behaviors in heterogeneous environments.

Contribution

It extends existing results by establishing new asymptotic behaviors of principal eigenvalues under large advection in heterogeneous settings for parabolic operators.

Findings

Asymptotic behaviors characterized for large advection

Extensions of prior elliptic and parabolic eigenvalue results

Analysis includes environments with spatial or temporal degeneracy

Abstract

We investigate the effects of advection on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions. Various asymptotic behaviors of the principal eigenvalues, when advection coefficient approaches infinity, are established in heterogeneous environments, where spatial or temporal degeneracy could occur in the advection term. Our findings partially extend the existing results in Chen-Lou [2008 Indiana Univ. Math. J.] and Peng-Zhou [2018 Indiana Univ. Math. J.] for elliptic operators and those in Peng-Zhao [2015 Calc. Var. Partial Diff.] for parabolic operators.