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

Abstract

In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -DΔϕ-2α\nabla m(x)\cdot \nablaϕ+V(x)ϕ=λϕ \ \hbox{ in }Ω, \end{equation} complemented by a general boundary condition including Dirichlet boundary condition and Robin boundary condition: $$\frac{\partial \ddot{I}•}{\partial n}+{\hat{I}}^2(x)\ddot{I}•=0on\partial \hat{I}\text{c◯},$$ where ${\hat{I}}^2\in C(\partial \hat{I}\text{c◯})$ allows to be positive, sign-changing or negative, and $n(x)$ is the unit exterior normal to $\partial \hat{I}\text{c◯}$ at $x$. The domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ is bounded and smooth, the constants $D>0$ and $\hat{I}\pm >0$ are, respectively, the diffusive and advection coefficients, and $m\in C^2(\bar{\hat{I}}\text{c◯}),V\in C(\bar{\hat{I}}\text{c◯})$ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient $D\to 0$ or $D\to \infty$. Our results, together with those of \cite{CL2,DF,Fr} where the Nuemann boundary case (i.e., ${\hat{I}}^2=0$ on $\partial \hat{I}\text{c◯}$) 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.