Asymptotic Analysis - Volume 43, issue 3

Authors: Albeverio, Sergio | Dobrokhotov, Sergey | Poteryakhin, Michael

Article Type: Research Article

Abstract: Smooth vector fields V in \[$\mathbb{R}$ n having a k-dimensional invariant torus are considered and the possibility of constructing the asymptotic eigenfunctions localized in the neighborhood of this torus is analyzed for the case of a corresponding small diffusion operator V·∇−εΔ, ε>0. The asymptotic stability property of the invariant torus is a sufficient condition for the existence of such eigenfunctions. Both the regular case (where the variational system is reducible) and the nonregular case (where the variational system is nonreducible) are considered.

Keywords: quasimodes, stable invariant tori, nonregular neighborhood, small diffusion, asymptotic stability

Citation: Asymptotic Analysis, vol. 43, no. 3, pp. 171-203, 2005

Authors: Matthies, Karsten

Article Type: Research Article

Abstract: We consider systems of semilinear elliptic equations on infinite cylinders with a nonlinear rapid periodic inhomogeneity in the unbounded direction. We transform the equation, such that the inhomogeneous term is exponentially small in the period of the inhomogeneity for bounded solutions. The results can be used to show that equilibrium solutions persist as periodic solutions with exponentially small modulation. The analytic tools of the paper include the dynamical systems approach to elliptic equations, averaging of exponential order for ordinary differential equations and extreme regularity (Gevrey classes).

Citation: Asymptotic Analysis, vol. 43, no. 3, pp. 205-232, 2005

Authors: Bonfoh, Ahmed

Article Type: Research Article

Abstract: We consider a singular perturbation of a generalized Cahn–Hilliard equation based on constitutive equations derived by M. Gurtin. Compared to the classical Cahn–Hilliard equation, these models take into account the work of internal microforces and the anisotropy of the material. We prove the existence of exponential attractors and their convergence with respect to the parameter of perturbation ε when it goes to zero.

Keywords: generalized Cahn–Hilliard equation, singular perturbation, exponential attractors, continuity of exponential attractors

Citation: Asymptotic Analysis, vol. 43, no. 3, pp. 233-247, 2005

Authors: Gwiazda, Piotr | Zatorska-Goldstein, Anna

Article Type: Research Article

Abstract: This note is concerned with an existence result for Leray–Lions quasilinear elliptic operator with discontinuous coefficients. The idea of the proof is based on compactness results for sequences of solutions to regularized problems obtained via compensated compactness, Young measures, and set-valued analysis tools.

Citation: Asymptotic Analysis, vol. 43, no. 3, pp. 249-265, 2005