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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: Coclite, Giuseppe Maria | di Ruvo, Lorenzo
Article Type: Research Article
Abstract: The wave propagation in dilatant granular materials is described by a nonlinear evolution equation of the fifth order deduced by Giovine–Oliveri in (Meccanica 30 (4) (1995 ) 341–357). In this paper, we study the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.
Keywords: Existence, Uniqueness, Stability, Wave propagation, Dilatant granular materials, Cauchy problem
DOI: 10.3233/ASY-241920
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-28, 2024
Authors: Shen, Liejun | Squassina, Marco
Article Type: Research Article
Abstract: We consider the existence of ground state solutions for a class of zero-mass Chern–Simons–Schrödinger systems − Δ u + A 0 u + ∑ j = 1 2 A j 2 u = f ( u ) − a ( x ) | u | p − 2 u , ∂ 1 A 2 − ∂ 2 A 1 = − 1 2 | u …| 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , ∂ 1 A 0 = A 2 | u | 2 , ∂ 2 A 0 = − A 1 | u | 2 , where a : R 2 → R + is an external potential, p ∈ ( 1 , 2 ) and f ∈ C ( R ) denotes the nonlinearity that fulfills the critical exponential growth in the Trudinger–Moser sense at infinity. By introducing an improvement of the version of Trudinger–Moser inequality approached in (J. Differential Equations 393 (2024 ) 204–237), we are able to investigate the existence of positive ground state solutions for the given system using variational method. Show more
Keywords: Zero-mass, Chern–Simons–Schrödinger system, Trudinger–Moser inequality, Critical exponential growth, Ground state solution, Variational method
DOI: 10.3233/ASY-241921
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-25, 2024
Authors: Ramos, A.J.A. | Rosário, L.G.M. | Campelo, A.D.S. | Freitas, M.M. | Martins, J.D.
Article Type: Research Article
Abstract: In the literature, we can find many studies that investigate the so-called truncated version of the Timoshenko beam system. In general, the truncated version eliminates the second spectrum of velocity and therefore does not require equal wave velocities to achieve exponential decay. However, the truncated system does not satisfy a Cauchy problem, which makes studying its qualitative properties more challenging. In this article, we present the truncated version of the laminated beam system. Our main results are the well-posedness of the problem using the classical Faedo-Galerkin method combined with a priori estimates and the exponential …decay of the energy functional without requiring equal wave velocities. Show more
Keywords: Laminated beams, truncated system, well-posedness, exponential decay
DOI: 10.3233/ASY-241918
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-23, 2024
Authors: Schäffner, M. | Schweizer, B.
Article Type: Research Article
Abstract: The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε , classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where …on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 . Show more
Keywords: Stochastic homogenization, wave equation
DOI: 10.3233/ASY-241923
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-39, 2024
Authors: Disconzi, Marcelo M. | Shao, Yuanzhen
Article Type: Research Article
Abstract: We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest.
Keywords: Strong hyperbolicity, first-order quasilinear systems, relativistic fluids
DOI: 10.3233/ASY-241919
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-22, 2024
Authors: Hajaiej, Hichem | Kumar, Rohit | Mukherjee, Tuhina | Song, Linjie
Article Type: Research Article
Abstract: This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type − ∂ x x u + ( − Δ ) y s 1 u + u − u 2 s 1 − 1 = κ α h ( x , y ) u α − 1 v β in R 2 …, − ∂ x x v + ( − Δ ) y s 2 v + v − v 2 s 2 − 1 = κ β h ( x , y ) u α v β − 1 in R 2 , u , v ⩾ 0 in R 2 , where s 1 , s 2 ∈ ( 0 , 1 ) , α , β > 1 , α + β ⩽ min { 2 s 1 , 2 s 2 } , and 2 s i = 2 ( 1 + s i ) 1 − s i , i = 1 , 2 . The existence of a ground state solution entirely depends on the behaviour of the parameter κ > 0 and on the function h . In this article, we prove that a ground state solution exists in the subcritical case if κ is large enough and h satisfies (H ). Further, if κ becomes very small, then there is no solution to our system. The study of the critical case, i.e., s 1 = s 2 = s , α + β = 2 s , is more complex, and the solution exists only for large κ and radial h satisfying (H1 ). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on h . Show more
Keywords: Mixed Schrödinger operator, system of PDEs in plane, variational methods, concentration-compactness, non-existence
DOI: 10.3233/ASY-241922
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-36, 2024
Authors: Ding, Jiajia | Estrada, Ricardo | Yang, Yunyun
Article Type: Research Article
Abstract: In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95 (1–2) (2015 ) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95 (1–2) (2015 ) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed.
Keywords: Distributions, asymptotic expansion, singular integral, Laplace formula, thick distributions
DOI: 10.3233/ASY-241924
Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-11, 2024
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