Asymptotic Analysis - Volume 61, issue 3-4

Authors: Castella, François | Hoffbeck, Jean-Philippe | Lagadeuc, Yvan

Article Type: Research Article

Abstract: We consider a spatially structured predator–prey model where fast migrations occur inside a given spatial domain, while slow predator–prey interactions prescribe the demographic evolution. The unknowns of our model are the numbers of predators and prey at each time t and each site x of the domain. In the idealized limit where migrations are infinitely fast, we show one can approximate the global dynamics using the mere two unknowns corresponding to the total number of preys and predators, irrespective of their respective spatial repartition. Besides, the error term induced by this approximation can be made exponentially small with respect to …the natural asymptotic parameter. In doing so, we completely characterize how migrations do modify both the qualitative and quantitative properties of the global demography. Our analysis relies on a convenient version of the central manifold theorem, in conjunction with a spectral gap estimate on the involved migration operator. Show more

Keywords: population dynamics, stability, central manifold, Perron–Frobenius, entropy estimate

DOI: 10.3233/ASY-2008-0905

Citation: Asymptotic Analysis, vol. 61, no. 3-4, pp. 125-175, 2009

Authors: Timofte, Aida

Article Type: Research Article

Abstract: We prove homogenization results for a class of rate-independent, nonlinear ferroelectric models. The PDE system defining the model is restated in terms of a stability condition and an energy balance law using an energy-storage functional and a dissipation functional. By the method of weak and strong two-scale convergence via periodic unfolding, we show that the solutions of the problem with periodicity converge to the energetic solution of the homogenized problem associated with the corresponding Γ-limits of the functionals. The main difficulties are the nonlinearity of the model, as well as the general form considered for the stored energy, which is …neither convex nor quadratic. Show more

Keywords: homogenization, ferroelectric, energetic formulation, periodic unfolding method

DOI: 10.3233/ASY-2008-0910

Citation: Asymptotic Analysis, vol. 61, no. 3-4, pp. 177-194, 2009

Authors: Giacomoni, J. | Prashanth, S. | Sreenadh, K.

Article Type: Research Article

Abstract: Let B1 be the unit open ball with center at the origin in RN , N≥2. We consider the following quasilinear problem depending on a real parameter λ>0: −ΔN u=λ f(u), u>0 in Ω, u=0 on ∂Ω,   (Pλ ) where f(t) is a nonlinearity that grows like etN/N−1 as t→∞ and behaves like tα , for some α∈(−∞, 0), as t→0+ . More precisely, we require f to satisfy assumptions (A1) and (A2) listed in Section 1. For such a general nonlinearity we show that if λ>0 is small enough, (Pλ ) admits at …least one weak solution (in the sense of distributions). We further study the question of uniqueness and multiplicity of solutions to (Pλ ) when Ω=B1 under additional structural conditions on f (see assumptions (A3)–(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)–(A5), we prove uniqueness of solution to (Pλ ) for all λ>0 small whereas under (A6), (A7) or (A8), we show multiplicity of solutions to (Pλ ) for all λ>0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition lim inf t→∞ h(t)teεt1/(N−1) =∞ ∀ε>0. Show more

Keywords: multiplicity, N-Laplace equation

DOI: 10.3233/ASY-2008-0911

Citation: Asymptotic Analysis, vol. 61, no. 3-4, pp. 195-227, 2009

Authors: Dupuy, Delphine | Orive, Rafael | Smaranda, Loredana

Article Type: Research Article

Abstract: In this paper we use the spectral method of Bloch waves to study the homogenization process of the Poisson equation in a periodically perforated domain, under homogeneous Dirichlet conditions on both exterior and interior boundaries, as the hole size goes to zero more rapidly than the micro-structure size. Using this method, we find the exact value of the critical hole size, which separates the different behaviors, where the classical strange term may or may not appear in the homogenized equation. This strange term is related to the asymptotic behavior of the first eigenvalue with respect to the hole radius.

Keywords: homogenization, perforated domains, Bloch waves, asymptotic behavior

DOI: 10.3233/ASY-2008-0912

Citation: Asymptotic Analysis, vol. 61, no. 3-4, pp. 229-250, 2009

Article Type: Other

Citation: Asymptotic Analysis, vol. 61, no. 3-4, pp. 251-251, 2009