Asymptotic Analysis - Volume 46, issue 3-4

Authors: Caillerie, D. | Raoult, A. | Sanchez-Palencia, E.

Article Type: Research Article

Abstract: We consider the system of equations of Koiter shell theory in a slightly simplified form, in the case when the limit (for small thickness) problem is hyperbolic, i.e., the principal curvatures of the middle surface are everywhere of opposite signs. Under loadings not belonging to the dual of the energy space of the limit problem, the energy of the solutions grows without limit as the thickness tends to zero and concentrates on internal or boundary layers. Two cases are considered, when the singular loadings are applied either along a non-characteristic or a characteristic curve. In both cases we define and …prove the convergence to the leading order in the layers. Show more

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 189-220, 2006

Authors: Caillerie, D. | Raoult, A. | Sanchez-Palencia, E.

Article Type: Research Article

Abstract: We consider the system of equations of Koiter shell theory in a slightly simplified form, in the case when the limit (for small thickness) problem is parabolic, i.e., the directions of the principal curvatures of the middle surface coincide everywhere. Under loadings not belonging to the dual of the energy space of the limit problem, the energy of the solutions grows without limit as the thickness tends to zero and concentrates on internal or boundary layers. Two cases are considered, when the singular loadings are applied either along a non-characteristic or a characteristic curve. In both cases we define and …prove the convergence to the leading order in the layers. Show more

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 221-249, 2006

Authors: Chepyzhov, V.V. | Pata, V.

Article Type: Research Article

Abstract: An abstract integrodifferential equation arising from linear viscoelasticity is considered. The stability properties of the related C0 -semigroup are discussed, in dependence on the form of the convolution (memory) kernel.

Keywords: linear viscoelasticity, memory kernels, $C_{0}$-semigroups, stability, exponential stability

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 251-273, 2006

Authors: Cîrstea, Florica Corina | Rădulescu, Vicenţiu

Article Type: Research Article

Abstract: We study the uniqueness and expansion properties of the positive solution of the logistic equation Δu+au=b(x)f(u) in a smooth bounded domain Ω, subject to the singular boundary condition u=+∞ on $\curpartial \varOmega $ . The absorption term f is a positive function satisfying the Keller–Osserman condition and such that the mapping f(u)/u is increasing on (0,+∞). We assume that b is non-negative, while the values of the real parameter a are related to an appropriate semilinear eigenvalue problem. Our analysis is based on the Karamata regular variation theory.

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 275-298, 2006

Authors: Destuynder, Philippe

Article Type: Research Article

Abstract: The goal of this paper is to derive a fluid-structure model from a three-dimensional formulation and using an asymptotic method. The mechanical model that we consider is a flow duct, a part of which is flexible. This part can be assimilated to a shell or a membrane structure and a smart device is located on it in order to control the vibrations induced by the flow inside the duct. This control model is known to be challenging and no realistic controllability result has been obtained as far as a three-dimensional model is considered. But it can be derived for one-dimensional …model. It is the reason why we are interested in this paper to analyze the links between the three-dimensional model and a limit one, assuming that the duct is quite long compared to its radius. The basic tool is a scaling method in the transverse section of the duct. Show more

Keywords: aeroacoustic, fluid-structure, asymptotic method, error analysis

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 299-323, 2006

Authors: Bartier, Jean-Philippe

Article Type: Research Article

Abstract: We study the equation ∂t u−Δu=up −μ |∇u|q ur in a general domain (bounded or not). Many results have been established in the case r=0, and we will generalize some properties. If q≥1 and q+r≥p, we will show that there is a strong connection between the finiteness of the inradius of Ω (or the Poincaré inequality) and the global existence of the solutions. More precisely if the inradius of Ω is finite, then the solutions u are global and if moreover μ is large enough, the solutions decay exponentially to zero. Conversely, if it is infinite, there always exists unbounded …solution. Other qualitative results are also obtained. Show more

Keywords: nonlinear parabolic equations, gradient term, finite time blow-up, global existence, inradius, Poincaré inequality

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 325-347, 2006

Authors: Hérau, Frédéric

Article Type: Research Article

Abstract: We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were inspired by former works on the Fokker–Planck equation and the main feature of this work is that they are relevant although the equation itself has no regularizing properties.

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 349-359, 2006

Article Type: Other

Citation: Asymptotic Analysis, vol. 46, no. 3-4, pp. 361-362, 2006