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

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.