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

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: Casado-Díaz, Juan

Article Type: Research Article

Abstract: The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in L N ( Ω ) , with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space H 0 1 ( Ω ) . However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates …in (J. Differential Equations 258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems. Show more

Keywords: Asymptotic behavior, elliptic problem, drift term, varying coefficients

DOI: 10.3233/ASY-241914

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-12, 2024

Authors: Guan, Minlan | Lai, Lizhen | Liu, Boxue | Qin, Dongdong

Article Type: Research Article

Abstract: In this paper, we study the following Hamilton–Choquard type elliptic system: − Δ u + u = ( I α ∗ F ( v ) ) f ( v ) , x ∈ R 2 , − Δ v + v = ( I β ∗ F ( u ) ) f ( u ) , x ∈ R 2 , where I α …and I β are Riesz potentials, f : R → R possessing critical exponential growth at infinity and F ( t ) = ∫ 0 t f ( s ) d s . Without the classic Ambrosetti–Rabinowitz condition and strictly monotonic condition on f , we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates. Show more

Keywords: Hamilton–Choquard elliptic system, Critical exponential growth, Ground state solution, Trudinger–Moser inequality

DOI: 10.3233/ASY-241916

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-31, 2024

Authors: Moutinho, Abdon

Article Type: Research Article

Abstract: In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ 6 model in dimension 1 + 1 . We construct a sequence of approximate solutions ( ϕ k ( v , t , x ) ) k ∈ N ⩾ 2 for this model to understand the effects of the collision in the movement of each soliton during a …large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ 6 model. Show more

Keywords: Kink, soliton, ϕ6 model, non-integrable model, scalar field, partial differential equation, ordinary differential equation, collision

DOI: 10.3233/ASY-241917

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-90, 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