Asymptotic Analysis - Volume 132, issue 3-4

Authors: Mi, Heilong | Zhang, Wen | Liao, Fangfang

Article Type: Research Article

Abstract: This paper is concerned with a class of fractional Schrödinger equation with Hardy potential ( − Δ ) s u + V ( x ) u − κ | x | 2 s u = f ( x , u ) , x ∈ R N , where s ∈ ( 0 , 1 ) and κ ⩾ 0 is a parameter. Under some suitable conditions on the potential V …and the nonlinearity f , we prove the existence of ground state solutions when the parameter κ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about κ . Finally, we are able to explore the asymptotic behavior of ground state solutions when κ tends to 0. Show more

Keywords: Fractional Schrödinger equation, Hardy potential, Strongly indefinite functional, Ground state solutions

DOI: 10.3233/ASY-221793

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 305-330, 2023

Authors: Stefanov, Plamen | Tindel, Samy

Article Type: Research Article

Abstract: We study the effect of additive noise to the inversion of FIOs associated to a diffeomorphic canonical relation. We use the microlocal defect measures to measure the power spectrum of the noise in the phase space and analyze how that power spectrum is transformed under the inversion. In general, white noise, for example, is mapped to noise depending on the position and on the direction. In particular, we compute the standard deviation, locally, of the noise added to the inversion as a function of the standard deviation of the noise added to the data. As an example, we study the …Radon transform in the plane in parallel and fan-beam coordinates, and present numerical examples. Show more

Keywords: Noise, Fourier Integral Operator, microlocal, inverse problem

DOI: 10.3233/ASY-221795

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 331-382, 2023

Authors: Falconi, Riccardo | Griso, Georges | Orlik, Julia

Article Type: Research Article

Abstract: This paper is focused on the asymptotic behavior of sequences of functions, whose partial derivatives estimates in one or more directions are highly contrasted with respect to the periodic parameter ε . In particular, a direct application for the homogenization of a homogeneous Dirichlet problem defined on an anisotropic structure is presented. In general, the obtained results can be applied to thin structures where the behavior is different according to the observed direction.

Keywords: Periodic unfolding method, homogenization, anisotropic Sobolev spaces, Dirichlet problem

DOI: 10.3233/ASY-221796

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 383-407, 2023

Authors: Chatzigeorgiou, Georgiana

Article Type: Research Article

Abstract: We study stochastic homogenization for linear elliptic equations in divergence form and focus on the recently developed theory of fluctuations. It has been observed that the fluctuations of averages of the solution are captured by the so-called standard homogenization commutator Ξ ε o . Our aim is to study how Ξ ε o (and its higher-order analogs) decorrelates on large scales when averaged on balls which are far enough. Taking advantage of its approximate locality, we give a quantitative characterization of this decorrelation in terms of …both the macroscopic scale and the distance between the balls showing that Ξ ε o inherits the correlation properties of the environment. Show more

Keywords: Stochastic homogenization, fluctuations, homogenization commutator, covariance estimate, higher-order theory

DOI: 10.3233/ASY-221797

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 409-426, 2023

Authors: Zhu, Rui | Tang, Xianhua

Article Type: Research Article

Abstract: We prove the existence and asymptotic behavior of solutions to the following problem: − Δ u + V ( x ) u − g ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where g ( x ) : = μ | …x | is called the Coulomb potential, g ( x ) : = β | x | 2 is called the Hardy potential (the inverse-square potential). μ , β > 0 are parameters, I α : R N ⟶ R is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When μ ∈ ( 0 , α ( N − 2 ) / 2 ( α + 1 ) ) and β ∈ ( 0 , α ( N − 2 ) 2 / 4 ( 2 + α ) ) , under some mild assumptions on V , we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature. Show more

Keywords: Choquard equation, ground state solution, Berestycki–Lions type conditions, Coulomb potential, Hardy potential

DOI: 10.3233/ASY-221798

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 427-450, 2023

Authors: Su, Yu | Liu, Zhisu

Article Type: Research Article

Abstract: In this paper, we are concerned with a class of Choquard equation with the lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. We emphasize that nonlinearities with doubly critical exponents are totally different from the well-known Berestycki–Lions-type ones. Working in a variational setting, we prove the existence, multiplicity and concentration of positive solutions for such equations when the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit.

Keywords: Choquard equation, doubly critical exponents, semi-classical state, variational method, Moser iteration

DOI: 10.3233/ASY-221799

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 451-493, 2023

Authors: Yan, Xingjie | Wang, Shubin | Yang, Xin-Guang | Zhang, Junzhao

Article Type: Research Article

Abstract: This paper is concerned with the long time behavior of solutions for a non-autonomous reaction-diffusion equations with anomalous diffusion. Under suitable assumptions on nonlinearity and external force, the global well-posedness has been studied. Then the pullback attractors in L 2 ( Ω ) and H 0 α ( Ω ) (0 < α < 1 ) have been achieved with a restriction on the growth order of nonlinearity as 2 ⩽ p ⩽ 2 ( n − α ) n − 2 α …. The results presented can be seen as the extension for classical theory of infinite dimensional dynamical system to the fractional diffusion equations. Show more

Keywords: Fractional Laplacian, pullback attractor, non-compactness measure

DOI: 10.3233/ASY-221800

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 495-517, 2023

Authors: Li, Lu

Article Type: Research Article

Abstract: This paper studies firstly the well-posedness and the asymptotic behavior of a Cahn–Hilliard–Oono type model, with cubic nonlinear terms, which is proposed for image segmentation. In particular, the existences of the global attractor and the exponential attractor have been proved, and it shows that the fractal dimension of the global attractor will tend to infinity as α → 0 . The difficulty here is that we no longer have the conservation of mass. Furthermore, this model with logarithmic nonlinear terms has been studied as well. One difficulty here is to make sure that the logarithmic terms can pass …to the limit under the standard Galerkin scheme. Another difficulty is to prove additional regularities on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. It eventually shows that the dimension of the global attractor is finite by proving the existence of the exponential attractor. Show more

Keywords: Image segmentation, Cahn–Hilliard–Oono equation, well-posedness, global attractor, exponential attractor, strict separation

DOI: 10.3233/ASY-221801

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 519-548, 2023

Authors: Nonato, C.A. | Raposo, C.A. | Feng, B. | Ramos, A.J.A.

Article Type: Research Article

Abstract: In this paper we consider a model of laminated beams combining viscoelastic damping and strong time-delayed damping. The global well-posedness is proved by using the theory of semigroups of linear operators. We prove the lack of exponential stability when the speed wave propagations are not equal. In fact, we show in this situation, that the system goes to zero polynomially with rate t − 1 / 2 . On the other hand, by constructing some suitable multipliers, we establish that the energy decays exponentially provided the equal-speed wave propagations hold.

Keywords: Laminated beams, Kelvin–Voigt damping, strong delay, exponential decay, polynomial decay

DOI: 10.3233/ASY-221802

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 549-574, 2023

Authors: Bader, Fakhrielddine | Bendahmane, Mostafa | Saad, Mazen | Talhouk, Raafat

Article Type: Research Article

Abstract: We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022 ) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate …temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result. Show more

Keywords: Tridomain model, reaction-diffusion system, homogenization theory, time-periodic unfolding method, gap junctions, cardiac electric field

DOI: 10.3233/ASY-221804

Citation: Asymptotic Analysis, vol. 132, no. 3-4, pp. 575-606, 2023