Asymptotic Analysis - Volume 125, issue 3-4

Authors: Nakamura, Makoto | Takeda, Hiroshi

Article Type: Research Article

Abstract: A semilinear diffusion equation is considered in the de Sitter spacetime with the spatially flat curvature. Global solutions for small initial data and their asymptotic behaviors in high order are obtained. Some effects of the spatial expansion and contraction are studied through the problem.

Keywords: Semilinear diffusion equation, global solution, asymptotic behavior, de Sitter spacetime

DOI: 10.3233/ASY-201652

Citation: Asymptotic Analysis, vol. 125, no. 3-4, pp. 203-245, 2021

Authors: Neukamm, Stefan | Varga, Mario | Waurick, Marcus

Article Type: Research Article

Abstract: Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the …approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation. Show more

Keywords: Periodic and stochastic homogenization, unfolding, abstract evolutionary equations, Maxwell’s equations

DOI: 10.3233/ASY-201654

Citation: Asymptotic Analysis, vol. 125, no. 3-4, pp. 247-287, 2021

Authors: Sauter, Stefan | Torres, Céline

Article Type: Research Article

Abstract: We study wave propagation phenomena modelled in the frequency domain by the Helmholtz equation in heterogeneous media with focus on media with discontinuous, highly oscillating wave speed. We restrict to problems with spherical symmetry and will derive explicit representations of the Green’s operator and stability estimates which are explicit in the frequency and the wave speed.

Keywords: Helmholtz equation, high frequency, heterogeneous media, stability estimates, Bessel functions

DOI: 10.3233/ASY-201657

Citation: Asymptotic Analysis, vol. 125, no. 3-4, pp. 289-325, 2021

Authors: Feireisl, Eduard | Lu, Yong | Sun, Yongzhong

Article Type: Research Article

Abstract: We consider a non–homogeneous incompressible and heat conducting fluid confined to a 3D domain perforated by tiny holes. The ratio of the diameter of the holes and their mutual distance is critical, the former being equal to ε 3 , the latter proportional to ε , where ε is a small parameter. We identify the asymptotic limit for ε → 0 , in which the momentum equation contains a friction term of Brinkman type determined uniquely by the viscosity and geometric properties of the perforation. Besides the inhomogeneity of the fluid, we …allow the viscosity and the heat conductivity coefficient to depend on the temperature, where the latter is determined via the Fourier law with homogenized (oscillatory) heat conductivity coefficient that is different for the fluid and the solid holes. To the best of our knowledge, this is the first result in the critical case for the inhomogenous heat–conducting fluid. Show more

Keywords: Non-homogeneous Navier–Stokes system, homogenization, heat–conducting fluid, incompressible fluid, Brinkman law

DOI: 10.3233/ASY-201658

Citation: Asymptotic Analysis, vol. 125, no. 3-4, pp. 327-346, 2021

Authors: Isozaki, Hiroshi | Korotyaev, Evgeny

Article Type: Research Article

Abstract: We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M = ( 0 , ∞ ) × Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove Asymptotics of counting function of resonances at large radius. The rotation radius is uniquely determined by its eigenvalues and resonances. There exists an algorithm to recover the rotation radius from its eigenvalues and resonances. …Asymptotics of counting function of resonances at large radius. The rotation radius is uniquely determined by its eigenvalues and resonances. There exists an algorithm to recover the rotation radius from its eigenvalues and resonances. The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces. Show more

Keywords: Inverse resonance scattering, rotationally symmetric manifolds, iso-resonance sets

DOI: 10.3233/ASY-201659

Citation: Asymptotic Analysis, vol. 125, no. 3-4, pp. 347-363, 2021

Authors: Hassan, Jamilu Hashim | Messaoudi, Salim A.

Article Type: Research Article

Abstract: In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type.

Keywords: Viscoelastic, general decay, relaxation function, variable exponent

DOI: 10.3233/ASY-201661

Citation: Asymptotic Analysis, vol. 125, no. 3-4, pp. 365-388, 2021

Authors: Chorfi, Nejmeddine

Article Type: Research Article

Abstract: The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

Keywords: Semi-linear heat equation, implicit Euler scheme, spectral elements discretization, mesh adaptivity, error indicators

DOI: 10.3233/ASY-201663

Citation: Asymptotic Analysis, vol. 125, no. 3-4, pp. 389-398, 2021