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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
Authors: Chakrabortty, Amartya | Griso, Georges | Orlik, Julia
Article Type: Research Article
Abstract: This paper focuses on the simultaneous homogenization and dimension reduction of periodic composite plates within the framework of non-linear elasticity. The composite plate in its reference (undeformed) configuration consists of a periodic perforated plate made of stiff material with holes filled by a soft matrix material. The structure is clamped on a cylindrical part. Two cases of asymptotic analysis are considered: one without pre-strain and the other with matrix pre-strain. In both cases, the total elastic energy is in the von-Kármán (vK) regime (ε 5 ). A new splitting of the displacements is introduced to …analyze the asymptotic behavior. The displacements are decomposed using the Kirchhoff–Love (KL) plate displacement decomposition. The use of a re-scaling unfolding operator allows for deriving the asymptotic behavior of the Green St. Venant’s strain tensor in terms of displacements. The limit homogenized energy is shown to be of vK type with linear elastic cell problems, established using the Γ-convergence. Additionally, it is shown that for isotropic homogenized material, our limit vK plate is orthotropic. The derived results have practical applications in the design and analysis of composite structures. Show more
Keywords: Homogenization, dimension reduction, unfolding operators, Γ-convergence, non-linear elasticity, von-Kármán plate, pre-strain
DOI: 10.3233/ASY-241896
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-56, 2024
Authors: Ganguly, Debdip | Gupta, Diksha | Sreenadh, K.
Article Type: Research Article
Abstract: We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N …denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞ , if N = 2 , λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ) . We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1 . Next, we prove that for a ( x ) ⩾ 1 , there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space. Show more
Keywords: Hyperbolic space, hyperbolic bubbles, Palais–Smale decomposition, semilinear elliptic problem
DOI: 10.3233/ASY-241895
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-29, 2024
Authors: Liu, Jitao | Wang, Shasha | Xu, Wen-Qing
Article Type: Research Article
Abstract: Recently, Niche [J. Differential Equations, 260 (2016), 4440–4453] established upper bounds on the decay rates of solutions to the 3D incompressible Navier–Stokes–Voigt equations in terms of the decay character r ∗ of the initial data in H 1 ( R 3 ) . Motivated by this work, we focus on characterizing the large-time behavior of all space-time derivatives of the solutions for the 2D case and establish upper bounds and lower bounds on their decay rates by making use of the decay character and Fourier splitting …methods. In particular, for the case − n 2 < r ∗ ⩽ 1 , we verify the optimality of the upper bounds, which is new to the best of our knowledge. Similar improved decay results are also true for the 3D case. Show more
Keywords: Incompressible Navier–Stokes–Voigt equations, decay characterization, Fourier splitting, large-time behavior
DOI: 10.3233/ASY-241900
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2024
Authors: Huo, Wenwen | Teng, Kaimin | Zhang, Chao
Article Type: Research Article
Abstract: We consider the Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.
Keywords: Navier–Stokes–Allen–Cahn, global well-posedness, time-decay, Fourier-splitting
DOI: 10.3233/ASY-241901
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-21, 2024
Authors: Kassan, Mouna | Carbou, Gilles | Jazar, Mustapha
Article Type: Research Article
Abstract: In this paper, we establish the existence of global-in-time weak solutions for the Landau–Lifschitz–Gilbert equation with magnetostriction in the case of mixed boundary conditions. From this model, we derive by asymptotic method a two-dimensional model for thin ferromagnetic plates taking into account magnetostrictive effects.
Keywords: Ferromagnetism, magnetostriction, weak solutions, thin plates
DOI: 10.3233/ASY-241899
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-29, 2024
Authors: Pelinovsky, Dmitry E. | Sobieszek, Szymon
Article Type: Research Article
Abstract: Ground state of the energy-critical Gross–Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end of this interval and diverges to infinity at the other end. We explore the shooting method in the limit of large norm to prove that the ground state is pointwise close to the Aubin–Talenti solution of the energy-critical wave equation in near field and to the confluent hypergeometric function in far field. The shooting method gives the precise dependence of the eigenvalue parameter versus …the supremum norm. Show more
Keywords: Gross–Pitaevskii equation, ground state, energy-critical case, shooting method
DOI: 10.3233/ASY-241897
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-29, 2024
Authors: Bouhoufani, Oulia | Messaoudi, Salim A. | Alahyane, Mohamed
Article Type: Research Article
Abstract: In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples.
Keywords: Biharmonic operator, Existence, Blow up, Coupled system, Variable exponent, Weak solution
DOI: 10.3233/ASY-231891
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-31, 2024
Authors: Barrué, Grégoire | Debussche, Arnaud | Tusseau, Maxime
Article Type: Research Article
Abstract: We prove that the stochastic Nonlinear Schrödinger (NLS) equation is the limit of NLS equation with random potential with vanishing correlation length. We generalize the perturbed test function method to the context of dispersive equations. Apart from the difficulty of working in infinite dimension, we treat the case of random perturbations which are not assumed uniformly bounded.
Keywords: Nonlinear Schrödinger equation, diffusion-approximation
DOI: 10.3233/ASY-241894
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-50, 2024
Authors: Badawi, Haidar | Alsayed, Hawraa
Article Type: Research Article
Abstract: In this paper, we consider a one dimensional thermoelastic Timoshenko system in which the heat flux is given by Cattaneo’s law and acts locally on the bending moment with a time delay. We prove its well-posedness, strong stability, and polynomial stability.
Keywords: Timoshenko system, Cattaneo’s law, strong stability, polynomial stability, frequency domain approach, time delay
DOI: 10.3233/ASY-231888
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-26, 2023
Authors: Khelifi, Hichem
Article Type: Research Article
Abstract: In this paper, we are interested in the existence and regularity of solutions for some anisotropic elliptic equations with Hardy potential and L m ( Ω ) data in appropriate anisotropic Sobolev spaces. The aim of this work is to get natural conditions to show the existence and regularity results for the solutions, that is related to an anisotropic Hardy inequality.
Keywords: Anisotropic elliptic problems, existence and regularity, Hardy potential, irregular data, Hardy inequality
DOI: 10.3233/ASY-231889
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-13, 2023
Authors: Zhang, Jiangwei | Xie, Zhe | Xie, Yongqin
Article Type: Research Article
Abstract: This paper aims to study the long-time behavior of nonclassical diffusion equation with memory and disturbance parameters on time-dependent space. By using the contractive process method on the family of time-dependent spaces and operator decomposition technique, the existence of pullback attractors is first proved. Then the upper semi-continuity of pullback attractors with respect to perturbation parameter ν in M t is obtained. It’s remarkable that the nonlinearity f satisfies the polynomial growth of arbitrary order.
Keywords: Nonclassical diffusion equation, arbitrary polynomial growth, pullback attractor, memory, upper semi-continuity
DOI: 10.3233/ASY-231887
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-23, 2023
Authors: Kitavtsev, Georgy | Taranets, Roman M.
Article Type: Research Article
Abstract: We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having non-homogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic H 1 -convergence of the solutions to a unique non-homogeneous positive steady state generally to hold.
Keywords: Asymptotic decay, steady states, porous medium, singular diffusion
DOI: 10.3233/ASY-231884
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-24, 2023
Authors: Sá Barreto, Antônio | Stefanov, Plamen
Article Type: Research Article
Abstract: We study the inverse problem of recovery a nonlinearity f ( t , x , u ) , which is compactly supported in x , in the semilinear wave equation u tt − Δ u + f ( t , x , u ) = 0 . We probe the medium with either complex or real-valued harmonic waves of wavelength ∼ h and amplitude ∼ 1 . They propagate in a regime where the nonlinearity affects the subprincipal but not the principal term, except for the zeroth harmonics. …We measure the transmitted wave when it exits supp x f . We show that one can recover f ( t , x , u ) when it is an odd function of u , and we can recover α ( x ) when f ( t , x , u ) = α ( x ) u 2 m . This is done in an explicit way as h → 0 . Show more
DOI: 10.3233/ASY-231890
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-42, 2024
Authors: Exner, Pavel | Kondej, Sylwia | Lotoreichik, Vladimir
Article Type: Research Article
Abstract: In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function R ∋ x ↦ d + ε f ( x ) , where d > 0 is a constant, ε > 0 is a small parameter, and f is a compactly supported continuous function. We prove that if ∫ R f d x > 0 , then the respective …Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small ε > 0 and we obtain the asymptotic expansion of this eigenvalue in the regime ε → 0 . An asymptotic expansion of the respective eigenfunction as ε → 0 is also obtained. In the case that ∫ R f d x < 0 we prove that the discrete spectrum is empty for all sufficiently small ε > 0 . In the critical case ∫ R f d x = 0 , we derive a sufficient condition for the existence of a unique bound state for all sufficiently small ε > 0 . Show more
Keywords: Schrödinger operators, strip-shaped potentials, discrete spectrum, weak deformation
DOI: 10.3233/ASY-241893
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-24, 2024
Authors: Al-Mahdi, Adel M.
Article Type: Research Article
Abstract: In this study, we consider a one-dimensional Timoshenko system with two damping terms in the context of the second frequency spectrum. One damping is viscoelastic with infinite memory, while the other is a non-linear frictional damping of variable exponent type. These damping terms are simultaneously and complementary acting on the shear force in the domain. We establish, for the first time to the best of our knowledge, explicit and general energy decay rates for this system with infinite memory. We use Sobolev embedding and the multiplier approach to get our decay results. These results generalize and improve some earlier related …results in the literature. Show more
Keywords: Timoshenko system, second frequency spectrum, multiplier method, infinite memory, exponential and polynomial decay, variable exponents
DOI: 10.3233/ASY-231892
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-33, 2024
Authors: Abdallaoui, Athmane | Kelleche, Abdelkarim
Article Type: Research Article
Abstract: In this paper, we start from a two dimensional transmission model problem in the framework of couple stress elasticity with voids which is defined in a fixed domain Ω − juxtaposed with a planar thin layer Ω + δ . We first derive a first approximation of Dirichlet-to-Neumann operator for the thin layer Ω + δ by using the techniques of asymptotic expansion with scaling, which allows us to approximate the transmission problem by a boundary value problem doesn’t take into …account any more the thin layer Ω + δ , called approximate impedance problem; after that, we prove an error estimate between the solution of the transmission problem and the solution of the approximate impedance problem. Show more
Keywords: Couple stress elasticity with voids, porous micropolar layer, Dirichlet-to-Neumann operator, impedance operator, asymptotic expansion
DOI: 10.3233/ASY-231886
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-21, 2023
Authors: Hatzizisis, Nicholas | Kamvissis, Spyridon
Article Type: Research Article
Abstract: In this paper we study the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, multi-humped, fairly smooth but not necessarily analytic potential decaying at infinity. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient.
Keywords: Integral equations, operator theory, complex analysis, inverse scattering, Jost solutions, Wentzel–Kramers–Brillouin approximation, Schrödinger equation
DOI: 10.3233/ASY-231885
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-67, 2023
Authors: Nika, Grigor | Muntean, Adrian
Article Type: Research Article
Abstract: We derive effective models for a heterogeneous second-gradient elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ , the intrinsic lengths of the constituents ℓ SG and ℓ chiral , and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓ SG , ℓ chiral , ℓ , and L we obtain either an …effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors’ structure rely on a suitable use of the periodic unfolding operator. Show more
Keywords: Second-gradient elasticity, scale-size effects, partial scale separation, chirality, multi-continuum homogenization
DOI: 10.3233/ASY-241902
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2024
Authors: Nguyen-Tien, Hoang
Article Type: Research Article
Abstract: We study the optimal convergence rate for the homogenization problem of convex Hamilton–Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of (Tran and Yu (2021 )), which means the optimal convergence rate is also O ( ε ) .
Keywords: Hamilton–Jacobi equations, homogenization, spatio-temporal periodic setting, optimal convergent rate, viscosity solutions
DOI: 10.3233/ASY-241898
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-16, 2024
Authors: Poblete, Felipe | Silva, Clessius | Viana, Arlúcio
Article Type: Research Article
Abstract: This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ …= 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β . Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions. Show more
Keywords: Nonlocal Navier–Stokes, Boussinesq system, PDEs in connection with fluid mechanics, fractional memory, self-similarity
DOI: 10.3233/ASY-241904
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-25, 2024
Authors: Liu, Chungen | Zhong, Yuyou | Zuo, Jiabin
Article Type: Research Article
Abstract: In this paper, we study a fractional Schrödinger–Poisson system with p -Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.
Keywords: Fractional Schrödinger–Poisson system, p-Laplacian, mountain pass lemma
DOI: 10.3233/ASY-241903
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-17, 2024
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