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In this article, we are concerned with the following eigenvalue problem of alinear 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 conditionincluding 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 ornegative, 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 advectioncoefficients, and $m\in C^2(\bar\Omega),\,V\in C(\bar\Omega)$ are givenfunctions.  We aim to investigate the asymptotic behavior of the principal eigenvalue ofthe above eigenvalue problem as the diffusive coefficient $D\to0$ or$D\to\infty$. Our results, together with those of \cite{CL2,DF,Fr} where theNuemann boundary case (i.e., $\beta=0$ on $\partial\Omega$) and Dirichletboundary case were studied, reveal the important effect of advection andboundary conditions on the asymptotic behavior of the principal eigenvalue. Wealso apply our results to a reaction-diffusion-advection equation which is usedto describe the evolution of a single species living in a heterogeneous streamenvironment and show some interesting behaviors of the species persistence andextinction caused by the buffer zone and small/large diffusion rate.

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