Asymptotic Analysis - Volume Pre-press, issue Pre-press

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. Pre-press, no. Pre-press, pp. 1-24, 2022

Authors: Dabrock, Nils | Knüttel, Sascha | Röger, Matthias

Article Type: Research Article

Abstract: We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.

Keywords: Willmore flow, phase-field model, diffuse interface, sharp interface limit

DOI: 10.3233/ASY-221810

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-42, 2022

Authors: Nandakumaran, A.K. | Sufian, Abu | Thazhathethil, Renjith

Article Type: Research Article

Abstract: In the present article, we study the homogenization of a second-order elliptic PDE with oscillating coefficients in two different domains, namely a standard rectangular domain with very general oscillations and a circular type oscillating domain. Further, we consider the source term in L 1 and hence the solutions are interpreted as renormalized solutions. In the first domain, oscillations are in horizontal directions, while that of the second one is in the angular direction. To take into account the type of oscillations, we have used two different types of unfolding operators and have studied the …asymptotic behavior of the renormalized solution of a second-order linear elliptic PDE with a source term in L 1 . In fact, we begin our study in oscillatory circular domain with oscillating coefficients and L 2 data which is also new in the literature. We also prove relevant strong convergence (corrector) results. We present the complete details in the context of circular domains, and sketch the proof in other domain. Show more

Keywords: Homogenization, periodic unfolding, oscillating boundary, circular oscillating domain, renormalized solution

DOI: 10.3233/ASY-221808

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-36, 2022

Authors: Appolloni, Luigi | Fiscella, Alessio | Secchi, Simone

Article Type: Research Article

Abstract: We consider a quasilinear partial differential equation governed by the p -Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.

Keywords: Fractional p-Kirchhoff equation, critical growth

DOI: 10.3233/ASY-221809

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-25, 2022

Authors: Shang, Bin | Zhang, Chao

Article Type: Research Article

Abstract: We establish a strong maximum principle for weak solutions of the mixed local and nonlocal p -Laplace equation − Δ p u + ( − Δ ) p s u = c ( x ) | u | p − 2 u in  Ω , where Ω ⊂ R N is an open set, p ∈ ( 1 , ∞ ) , s ∈ ( 0 , 1 ) …and c ∈ C ( Ω ‾ ) . Show more

Keywords: Strong maximum principle, weak solutions, viscosity solutions, mixed local and nonlocal p-Laplace operator

DOI: 10.3233/ASY-221803

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-12, 2022

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. Pre-press, no. Pre-press, pp. 1-23, 2022

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. Pre-press, no. Pre-press, pp. 1-32, 2022

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. Pre-press, no. Pre-press, pp. 1-26, 2022

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. Pre-press, no. Pre-press, pp. 1-43, 2022

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. Pre-press, no. Pre-press, pp. 1-30, 2022