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

2020/02/04 by Shuang Liu, Liu, Shuang, Yuan Lou +5 · 2 citations
Computer Science · Mathematics · Engineering · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2002.01357

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.

Cited by

• Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection
• The principal eigenvalue of an age-structured operator with diffusion and advection: qualitative analysis and an application

Related

• Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator I: Large advection
• On principal eigenvalues of linear time-periodic parabolic systems: symmetric mutation case
• Reaction-Diffusion Problems on Time-Periodic Domains
• On the asimptotic behavior of solutions of the Cauchy problem for parabolic equations with time periodic coefficients
• Principal eigenvalue for some elliptic operators with large drift: Neumann boundary conditions