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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: 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: 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
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: Deng, Ting | Squassina, Marco | Zhang, Jianjun | Zhong, Xuexiu
Article Type: Research Article
Abstract: We are concerned with solutions of the following quasilinear Schrödinger equations − div ( φ 2 ( u ) ∇ u ) + φ ( u ) φ ′ ( u ) | ∇ u | 2 + λ u = f ( u ) , x ∈ R N with prescribed mass ∫ R N u 2 d x = c , …where N ⩾ 3 , c > 0 , λ ∈ R appears as the Lagrange multiplier and φ ∈ C 1 ( R , R + ) . The nonlinearity f ∈ C ( R , R ) is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case. Show more
Keywords: Quasilinear Schrödinger equations, normalized solutions, mass critical exponent
DOI: 10.3233/ASY-241908
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-20, 2024
Authors: Chipot, Michel
Article Type: Research Article
Abstract: The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p -Laplacian type when the size of the domain goes to infinity in different directions.
Keywords: Anisotropic operators, nonlinear elliptic operators, pseudo p-Laplacian, asymptotic behaviour, cylinder like domains
DOI: 10.3233/ASY-241906
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2024
Authors: Rawat, Rama | Roy, Haripada | Roy, Prosenjit
Article Type: Research Article
Abstract: The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p -Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85 (3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.
Keywords: p-Laplacian, uniform elliptcity, Poincaré inequality, Krasnoselskii’s genus
DOI: 10.3233/ASY-241907
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-33, 2024
Authors: Kundu, A. | Mahato, H.S.
Article Type: Research Article
Abstract: We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.
Keywords: Diffusion–reaction–precipitation equations, optimal control problem, homogenisation, periodic porous medium, existence of solution, asymptotic expansion, two-scale convergence
DOI: 10.3233/ASY-241905
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-33, 2024
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