Asymptotic Behavior of the Principal Eigenvalue of a Linear Second Order Elliptic Operator with Small/Large Diffusion Coefficient

Abstract

In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator:complemented by a general boundary condition including Dirichlet boundary condition and Robin boundary condition:where β ∈ C(∂Ω) allows to be positive, sign-changing or negative, and n(x) is the unit exterior normal to ∂Ω at x. The domain Ω ⊂ R N is bounded and smooth, the constants D > 0 and α > 0 are, respectively, the diffusive and advection coefficients, and m ∈ C 2 (Ω), V ∈ C(Ω) are given functions.We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient D → 0 or D → ∞. Our results, together with those of [4,5,10] where the Nuemann boundary case (i.e., β = 0 on ∂Ω) 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.

Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator

“…For this simple-looking ODE eigenvalue problem, identifying the sign of λ 1 is not yet an easy task. In the recent works [5,6,16,21,23], the asymptotics of λ 1 for large α or small D in this case has been extensively studied. However, when V or/and ∂ x m depend both on the spatio-temporal variables, much less has been known for the behaviors of λ 1 ; the mathematical difficulty mainly comes from the lack of variational structure for problem (1.1).…”

“…Therein, the limit of the principal eigenvalue λ(D) as D → 0 plays a pivotal role in studying the large time behavior of the trajectories of stochastic systems; see also [7,10]. Recently the asymptotic behavior of λ(D) for problem (1.3) has been considered in [6] for general bounded domains, and their result in particular implies We refer to [21] for recent progress on problem (1.3) under general boundary conditions. Theorem 1.1 indicates that the limit of λ(D) relies upon the set of critical points of function m in the elliptic scenario.…”

“…(i) We note that λ(D) for (1.3) in the elliptic situation can be characterized by variational formulation [5,6,21,23]. In contrast, the time-periodic parabolic problem (1.1) has no variational formulations.…”

Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate

“…There have been a large literature about the effect of advection and diffusion on the principal eigenvalue of a linear second order elliptic operator [7,8,43,45]. Next we refer to [10] for some recent works on the asymptotic behaviors of the principal eigenvalue λ 1 (d S , d I , q) as advection rate tends to infinity or the diffusion coefficient tends to zero, which are relevant for the existence of EE.…”

“…Clearly, it follows from (48) thatN > 0. Furthermore, one can see from (43) and the assertion of Step 5 thatN < N . This completes the proof.…”