Asymptotic Analysis - Volume 120, issue 1-2

Authors: Conti, Monica | Pata, Vittorino | Quintanilla, Ramon

Article Type: Research Article

Abstract: In this paper, we consider a thermoelastic model where heat conduction is described by the history dependent version of the Moore–Gibson–Thompson equation, arising via the introduction of a relaxation parameter in the Green-Naghdi type III theory. The well-posedness of the resulting integro-differential system is discussed. In the one-dimensional case, the exponential decay of the energy is proved.

Keywords: Moore–Gibson–Thompson equation, relaxation parameter, memory kernel, thermoelasticity, solution semigroup, exponential stability

DOI: 10.3233/ASY-191576

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 1-21, 2020

Authors: Ciuperca, I. | Jai, M. | Tello, J.I.

Article Type: Research Article

Abstract: In this article we study a lubricated system consisting on a slider moving over a smooth surface and a known external force (the load) applied upon the slider. The slider moves at constant velocity and close proximity to the surface and the gap is filled by an incompressible fluid (the lubricant). At the equilibrium, the position of the slider presents one degree of freedom to be determined by the balance of forces acting on the system: the load and the total force exerted by the pressure of the lubricant. The pressure distribution is described by a variational inequality of elliptic …type known as Swift–Stieber model and based on Reynolds equation. The distance h between the surfaces in a two dimensional domain Ω is given by h η ( x 1 , x 2 , y ) = h 0 ( x 1 , x 2 ) + h 1 ( y ) + η , ( x 1 , x 2 ) ∈ Ω , y ∈ [ 0 , 1 ] where h 0 ( x 1 , x 2 ) ∼ | x 1 | α for α > 0 and h 1 ( y ) ∼ | y − y 0 | β for y being the homogenization variable. The main result of the article quantify the influence of the roughness in the load capacity of the mechanism in the following way: If  α < 3 γ for  0 < γ ⩽ 2 α < min { 1 γ − 2 , 3 γ } for  γ > 2 then, the mechanism presents finite load capacity, i.e. lim η → 0 ∫ Ω p η < ∞ . Infinite load capacity is obtained for γ > 1 and α > 2 / ( γ − 1 ) . A one dimensional particular case is given for γ > 3 / 2 with infinite load capacity. Show more

Keywords: Lubrication, Reynolds variational inequality, homogenization, inverse problem

DOI: 10.3233/ASY-191577

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 23-40, 2020

Authors: Colli, Pierluigi | Gilardi, Gianni | Sprekels, Jürgen

Article Type: Research Article

Abstract: In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn–Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Meth. Biomed. Eng. 28 (2012 ), 3–24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn–Hilliard equation for the tumor cell fraction …φ , coupled to a reaction–diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases. Show more

Keywords: Fractional operators, Cahn–Hilliard systems, well-posedness, regularity of solutions, tumor growth models, asymptotic analysis

DOI: 10.3233/ASY-191578

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 41-72, 2020

Authors: Muda, Yuslenita | Akbar, Fiki T. | Kusdiantara, Rudy | Gunara, Bobby E. | Susanto, Hadi

Article Type: Research Article

Abstract: We consider a discrete nonlinear Klein–Gordon equation with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein–Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.

Keywords: Discrete nonlinear Schrödinger equation, discrete Klein–Gordon equation, small-amplitude approximation, discrete breather, discrete soliton

DOI: 10.3233/ASY-191579

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 73-86, 2020

Authors: Monticelli, Dario D. | Punzo, Fabio | Squassina, Marco

Article Type: Research Article

Abstract: We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

Keywords: Nonexistence of solutions, hyperbolic problems, wave equation

DOI: 10.3233/ASY-191580

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 87-101, 2020

Authors: Baus, Franziska | Klar, Axel | Marheineke, Nicole | Wegener, Raimund

Article Type: Research Article

Abstract: This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e., low-Mach-number–slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply …a scheme that is based on a Gauss–Legendre collocation in space and an α -method in time. Show more

Keywords: Dynamic elastic Cosserat rod, Kirchhoff beam, low-Mach-number–slenderness limit, asymptotic analysis, asymptotic-preserving scheme

DOI: 10.3233/ASY-191581

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 103-121, 2020

Authors: Mohammed, Mogtaba | Ahmed, Noor

Article Type: Research Article

Abstract: In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [Cioranescu, Donato and Zaki in Port. Math. (N.S.) 63 (2006 ), 467–496]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the …boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems. Show more

Keywords: Periodic unfolding, SPDEs, Robin problem, perforated domains, probabilistic compactness results

DOI: 10.3233/ASY-191582

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 123-149, 2020

Authors: Giga, Yoshikazu | Liu, Qing | Mitake, Hiroyoshi

Article Type: Research Article

Abstract: We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo’s time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.

Keywords: Approximation to solutions, Caputo’s time fractional derivatives, second order fully nonlinear equations, viscosity solutions

DOI: 10.3233/ASY-191583

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 151-162, 2020

Authors: Guo, Jong-Shenq | Poh, Amy Ai Ling | Shimojo, Masahiko

Article Type: Research Article

Abstract: In this paper, we study an SIR epidemic model with nonlocal dispersal. We study the case with vital dynamics so that a renewal of the susceptible individuals is taken into account. We characterize the asymptotic spreading speed to estimate how fast the disease under consideration spreads. Due to the lack of comparison principle for the SIR model, our proof is based on a delicate analysis of related problems with nonlocal scalar equations.

Keywords: Epidemic model, spreading speed, nonlocal dispersal

DOI: 10.3233/ASY-191584

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 163-174, 2020

Authors: Yoshida, Naoya

Article Type: Research Article

Abstract: We study the eigenvalues of the two-dimensional Schrödinger operator with a large constant magnetic field perturbed by a decaying scalar potential. For each Landau level, we give the precise asymptotic distribution of eigenvalues created by the minimum, maximum and the closed energy curve of the potential. Normal form reduction, WKB construction and pseudodifferential calculus are applied to the effective Hamiltonian.

Keywords: Landau level, effective Hamiltonian, WKB construction, pseudodifferential calculus, Bohr–Sommerfeld quantization condition

DOI: 10.3233/ASY-191585

Citation: Asymptotic Analysis, vol. 120, no. 1-2, pp. 175-197, 2020