Asymptotic Analysis - Volume 94, issue 3-4

Authors: Bai, Lihong | Zhang, Fanghong

Article Type: Research Article

Abstract: In this article, the existence of a uniform attractor is proved for the multi-valued process generated by non-autonomous nonclassical diffusion equations (NDEs) with delays in unbounded domain without uniqueness of solutions, when the external force g ( x , t ) belongs to the space L C ∗ 2 ( R , L 2 ( R N ) ) and is a translation bounded function, but is not a translation compact function.

Keywords: uniform attractors, multi-valued process, nonclassical diffusion equations with delay, unbounded domain

DOI: 10.3233/ASY-151299

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 187-210, 2015

Authors: de Maio, Umberto | Faella, Luisa | Soueid, Salwa

Article Type: Research Article

Abstract: In this paper we study the asymptotic behavior of the solutions of time dependent micromagnetism problem in a multi-domain consisting of two joined ferromagnetic thin films. We distinguish different regimes depending on the limit of the ratio between the small thickness of the two films.

Keywords: micromagnetics, variational problem, thin films, Landau–Lifshitz equation

DOI: 10.3233/ASY-151311

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 211-240, 2015

Authors: Lussardi, Luca

Article Type: Research Article

Abstract: We investigate, using the framework of Γ -convergence, a phase-field model proposed in [Proc. R. Soc. A 465 (2009), 1337–1359] for strongly anisotropic systems; in particular, we prove a full Γ -convergence result for an anisotropic Modica–Mortola-type energy.

Keywords: phase-field, anisotropy, Γ-convergence

DOI: 10.3233/ASY-151312

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 241-254, 2015

Authors: Griso, G. | Villanueva-Pesqueira, Manuel

Article Type: Research Article

Abstract: In this paper, we consider rods whose thickness vary linearly between ϵ and ϵ 2 . Our aim is to study the asymptotic behavior of these rods in the framework of the linear elasticity. We use a decomposition method of the displacement fields of the form u = U e + u ¯ , where U e stands for the translation-rotations of the cross-sections and u ¯ is related to their deformations. We establish a priori …estimates. Passing to the limit in a fixed domain gives the problems satisfied by the bending, the stretching and the torsion limit fields which are ordinary differential equations depending on weights. Show more

Keywords: linear elasticity, rods

DOI: 10.3233/ASY-151313

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 255-291, 2015

Authors: Quittner, Pavol

Article Type: Research Article

Abstract: We consider an initial value problem for a singular second-order differential equation stemming from an optimal control problem in financial economics and possessing infinitely many solutions. If one looks for these solutions in the form of a power series then a formal approach leads to a series whose radius of convergence is zero (for some values of the parameters of the problem). We prove that each partial sum of this series is a good approximation of all the solutions close to the singularity: This solves an open question posed by P. Brunovský, A. Černý and M. Winkler. We also obtain uniqueness and comparison …results for the solutions satisfying an additional boundary condition and use these results in the study of the asymptotics of the difference of two solutions of the initial value problem. Show more

Keywords: singular differential equation, asymptotic expansion

DOI: 10.3233/ASY-151314

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 293-308, 2015

Authors: Le Meur, Hervé V.J.

Article Type: Research Article

Abstract: In this article, we derive a viscous Boussinesq system for surface water waves from Navier–Stokes equations for non-vanishing initial conditions. The relevance of such initial conditions is proved. We use neither the irrotationality assumption, nor the Zakharov–Craig–Sulem formulation. During the derivation, we find the bottom shear stress and also the decay rate for shallow water. In order to justify our derivation, we derive the viscous Korteweg–de Vries equation from our viscous Boussinesq system and compare it to the ones found in the bibliography. We also extend the system to the 3D geometry.

Keywords: water waves, shallow water, Boussinesq system, viscosity, KdV equation

DOI: 10.3233/ASY-151315

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 309-345, 2015

Authors: Figueiredo, Giovany M. | Molica Bisci, Giovanni | Servadei, Raffaella

Article Type: Research Article

Abstract: In this paper we study a highly nonlocal problem involving a fractional operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Precisely, we consider the following nonlocal problem − M ( ∥ u ∥ X 0 2 ) L K u = f ( x , u ) ( ∫ Ω ( ∫ 0 u ( x ) f ( x , τ ) d τ …) d x ) r in  Ω , u = 0 in  R n ∖ Ω , where r ⩾ 0 , s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, f : Ω ‾ × R → R and M : [ 0 , + ∞ ) → [ 0 , + ∞ ) are functions verifying suitable conditions and L K is the integrodifferential operator defined as follows L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , where the kernel K satisfies some natural conditions. By working in the fractional Sobolev space X 0 , which encodes Dirichlet homogeneous boundary conditions, and exploiting the genus theory introduced by Krasnoselskii, we derive the existence of infinitely many weak solutions for this problem. Show more

Keywords: Kirchhoff-type equations, fractional Laplacian, nonlocal problems, variational methods, critical point theory, Krasnoselskii’s genus

DOI: 10.3233/ASY-151316

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 347-361, 2015

Authors: Castro, Hernán

Article Type: Research Article

Abstract: We study the semi-linear Sturm–Liouville equation − ( x 2 α u ′ ) ′ = λ u + | u | p − 1 u in  ( 0 , 1 ) , u ( 1 ) = 0 , where α ⩾ 1 , p > 1 and λ are real parameters. We …prove that all non-trivial solutions are oscillatory and unbounded as x approaches 0. Moreover, there exist γ > 0 and δ > 0 such that any solution u ( x ) resembles x − γ η ( x − δ ) near the origin, where η is a non-trivial periodic solution to the Emden–Fowler equation δ 2 η ″ + | η | p − 1 η = 0 in [ 0 , ∞ ) . Show more

Keywords: oscillatory solutions, singular Sturm–Liouville equation, Emden–Fowler equation

DOI: 10.3233/ASY-151318

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 363-373, 2015

Article Type: Other

Citation: Asymptotic Analysis, vol. 94, no. 3-4, pp. 375-376, 2015