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

Abstract

We investigate the effect of small diffusion on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions in one dimensional space. The asymptotic behaviors of the principal eigenvalues, as the diffusion coefficients tend to zero, are established for non-degenerate and degenerate spatial-temporally varying environments. A new finding is the dependence of these asymptotic behaviors on the periodic solutions of a specific ordinary differential equation induced by the drift. The proofs are based upon delicate constructions of super/sub-solutions and the applications of comparison principles.

A functional approach towards eigenvalue problems associated with incompressible flow

“…A similar result to Theorem 1.1 was established in our work [28], which is associated with the periodic-parabolic eigenvalue problem…”

“…Such problem remains an important area of active research [24,25,31,33,35,36], with particular interest on the dependence of the principal eigenvalue λ 1 (τ ) on frequency τ . It is shown in [28] that λ 1 (τ ) is non-decreasing in τ > 0, and more precisely, (i) If c(x, t) = 1 0 c(x, s) ds + g(t) for some 1-periodic function g(t), then λ 1 (τ ) is constant for τ > 0; (ii) Otherwise ∂λ1 ∂τ (τ ) > 0 for every τ > 0. Identifying the connections between problems (3) and (33) is an interesting subject.…”

“…Recalling the boundary conditions of ϕ A in (28), the Hopf boundary lemma implies that ϕ A is a constant and thus ∇ϕ A = 0. Returning to (28), ∇ϕ A = 0 reads ∇ϕ A = 0 at the point λ 0 .…”

“…Proceeding to differentiate (28) in λ gives λ) in Ω, [aϕ A ] · n = 0 on ∂Ω, ϕ A (0, y) = ϕ A (L, y) and ∇ϕ A (0, y) = ∇ϕ A (L, y) in Ω Y .…”

Influence of mutations in phenotypically-structured populations in time periodic environment

“…In the present paper, one of our aims is to show that, in time-periodic media, it is sometimes more advantageous for the population to keep a positive mutation rate D > 0, in the sense that it could give a larger mean population than D 0. As an application of our results, we for example derive from [25] that time-heterogeneity always increases the mean population, while such a positive dependence is not clear when D 0.…”