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

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

Authors: Kinra, Kush | Mohan, Manil T.

Article Type: Research Article

Abstract: In this work, we consider the two-dimensional Oldroyd model for the non-Newtonian fluid flows (viscoelastic fluid) in Poincaré domains (bounded or unbounded) and study their asymptotic behavior. We establish the existence of a global attractor in Poincaré domains using asymptotic compactness property. Since the high regularity of solutions is not easy to establish, we prove the asymptotic compactness of the solution operator by applying Kuratowski’s measure of noncompactness, which relies on uniform-tail estimates and the flattening property of the solution. Finally, the estimates for the Hausdorff as well as fractal dimensions of global attractors are also obtained.

Keywords: Oldroyd fluids, Global attractors, Kuratowski’s measure of noncompactness, uniform-tail estimates, flattening property, fractal and Hausdorff dimensions

DOI: 10.3233/ASY-241932

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2024

Authors: Giacomoni, J. | Nidhi, Nidhi | Sreenadh, K.

Article Type: Research Article

Abstract: In this paper we study the existence and regularity results of normalized solutions to the following critical growth Choquard equation with mixed diffusion type operators: − Δ u + ( − Δ ) s u = λ u + g ( u ) + ( I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ − 2 u in  R N , ∫ R …N | u | 2 d x = τ 2 , where N ⩾ 3 , τ > 0 , I α is the Riesz potential of order α ∈ ( 0 , N ) , ( − Δ ) s is the fractional laplacian operator, 2 α ∗ = N + α N − 2 is the critical exponent with respect to the Hardy Littlewood Sobolev inequality, λ appears as a Lagrange multiplier and g is a real valued function satisfying some L 2 -supercritical conditions. Show more

Keywords: Normalized solutions, Choquard equation, critical growth, local and nonlocal operator, L2-supercritical growth, existence results, Sobolev regularity

DOI: 10.3233/ASY-241933

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-34, 2024

Authors: Hernández-Santamaría, Víctor | Peña-García, Alberto

Article Type: Research Article

Abstract: The shadow limit is a versatile tool used to study the reduction of reaction-diffusion systems into simpler PDE-ODE models by letting one of the diffusion coefficients tend to infinity. This reduction has been used to understand different qualitative properties and their interplay between the original model and its reduced version. The aim of this work is to extend previous results about the controllability of linear reaction-diffusion equations and how this property is inherited by the corresponding shadow model. Defining a suitable class of nonlinearities and improving some uniform Carleman estimates, we extend the results to the semilinear case and prove …that the original model is null-controllable and that the shadow limit preserves this important feature. Show more

Keywords: Shadow limit, semilinear reaction-diffusion equations, uniform null-controllability, Carleman estimates

DOI: 10.3233/ASY-241930

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-39, 2024

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: Li, Qingyue | Cheng, Xiyou | Yang, Lu

Article Type: Research Article

Abstract: In this paper, we are concerned with the following magnetic nonlinear equation of Kirchhoff type with critical exponential growth and an indefinite potential in R 2 m ( ∫ R 2 [ | 1 i ∇ u − A ( x ) u | 2 + V ( x ) | u | 2 ] d x ) [ ( 1 i ∇ − A ( x ) ) …2 u + V ( x ) u ] = B ( x ) f ( | u | 2 ) u , where u ∈ H 1 ( R 2 , C ) , m is a Kirchhoff type function, V : R 2 → R and A : R 2 → R 2 represent locally bounded potentials, while B denotes locally bounded and f exhibits critical exponential growth. By employing variational methods and utilizing the modified Trudinger–Moser inequality, we get ground state solutions or nontrivial solutions for the above equation. Furthermore, in the special case where m is a constant equal to 1, the equation is reduced to the following magnetic nonlinear Schrödinger equation, ( 1 i ∇ − A ( x ) ) 2 u + V ( x ) u = B ( x ) f ( | u | 2 ) u in  R 2 . Applying analogous methods, we can also establish the existence of ground state solutions or nontrivial solutions to this equation. Show more

Keywords: Kirchhoff–Schrödinger equation, magnetic field, indefinite potential, minimization method, Nehari manifold, Trudinger–Moser inequality

DOI: 10.3233/ASY-241929

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-36, 2024

Authors: Shi, Peilin | Dong, Lingzhen

Article Type: Research Article

Abstract: We consider a system of stochastic differential equations, which is established by introducing white noises into a periodic and ratio-dependent food-chain system of three-species. We investigate the asymptotic behaviors for such a system, and obtain the sufficient conditions for its extinction and persistence with the help of Itô’s formula. We discuss the existence of a nontrivial positive periodic solution based on Has’minskii theory of periodic solution. Further, we study a ratio-dependent food-chain system of three-species including not only white noise but also telephone noise. For such a model, we give the sufficient conditions to guarantee the existence of a unique …ergodic stationary distribution. Show more

Keywords: Ratio-dependent food-chain systems, Itô’s formula, extinction and persistence, stochastic periodic solution, stationary distribution

DOI: 10.3233/ASY-241927

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-20, 2024