Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with small/large diffusion coefficient and its application

TL;DR

This paper investigates how the principal eigenvalue of a second order elliptic operator behaves asymptotically as the diffusion coefficient approaches zero or infinity, considering various boundary conditions and applications to ecological models.

Contribution

It provides new insights into the asymptotic behavior of the principal eigenvalue under different boundary conditions and diffusion limits, extending previous studies.

Findings

Eigenvalue behavior varies significantly with boundary conditions.

Advection and boundary conditions critically influence eigenvalue asymptotics.

Application to ecological models shows effects on species persistence and extinction.

Abstract

In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \ \hbox{ in }\Omega, \end{equation} complemented by a general boundary condition including Dirichlet boundary condition and Robin boundary condition: $$ \frac{\partial\phi}{\partial n}+\beta(x)\phi=0 \ \ \hbox{ on }\partial\Omega, $$ where $\beta\in C(\partial\Omega)$ allows to be positive, sign-changing or negative, and $n(x)$ is the unit exterior normal to $\partial\Omega$ at $x$. The domain $\Omega\subset\mathbb{R}^N$ is bounded and smooth, the constants $D>0$ and $\alpha>0$ are, respectively, the diffusive and advection coefficients, and $m\in C^2(\bar\Omega),\,V\in C(\bar\Omega)$ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient $D\to0$ or $D\to\infty$. Our results, together with those of \cite{CL2,DF,Fr} where the Nuemann boundary case (i.e., $\beta=0$ on $\partial\Omega$) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue. We also apply our results to a reaction-diffusion-advection equation which is used to describe the evolution of a single species living in a heterogeneous stream environment and show some interesting behaviors of the species persistence and extinction caused by the buffer zone and small/large diffusion rate.