Effects of large degenerate advection and boundary conditions on the principal eigenvalue and its eigenfunction of a linear second order elliptic operator

TL;DR

This paper investigates how large degenerate advection and boundary conditions influence the principal eigenvalue and eigenfunction of a linear second-order elliptic operator, revealing new effects and improving existing results in one dimension.

Contribution

It provides a comprehensive analysis of the asymptotic behavior of the principal eigenvalue under degenerate advection and various boundary conditions, advancing understanding in one-dimensional cases.

Findings

Asymptotic limits depend on the degeneracy of the advection term.

Boundary conditions significantly affect the eigenvalue behavior.

New fundamental effects of degenerate advection are identified.

Abstract

In this article, we study, as the coefficient $s\to\infty$, the asymptotic behavior of the principal eigenvalue of $$-\varphi''(x)-2sm'(x)\varphi'(x)+c(x)\varphi(x)=\lambda_s\varphi(x),\ \ 0<x<1,$$ supplemented by different boundary conditions. This problem is relevant to nonlinear propagation phenomena in reaction-diffusion equations. The main point is that the advection (or drift) term $m$ allows natural degeneracy. For instance, $m$ could be constant on $[a,b]\subset[0,1]$. Depending on the behavior of $m$ near the neighbourhood of the end points $a,\,b$, the limiting value could be the principal eigenvalue of $$-\varphi''(x)+c(x)\varphi(x)=\lambda\varphi(x),\ \ a<x<b,$$ coupled with Dirichlet or Newmann boundary condition. A complete understanding of the limiting behavior of the principal eigenvalue and its eigenfunction is obtained, and new fundamental effects of large degenerate advection and boundary conditions on the principal eigenvalue and the principal eigenfunction are revealed. In one space dimension, the results in the existing literature are substantially improved.