Asymptotic Analysis - Volume 129, issue 3-4

Authors: Bostan, Mihaï

Article Type: Research Article

Abstract: The subject matter of this work concerns the propagation of the electro-magnetic fields through strongly anisotropic media, in the three dimensional setting. We concentrate on the asymptotic behavior for the solutions of the Maxwell equations when the electric permittivity tensor is strongly anisotropic. We derive limit models and prove their well-posedness. We appeal to the variational framework and study the propagation speed of the solutions. We prove that almost all the electro-magnetic energy concentrates inside the propagation cone of the limit model.

Keywords: Maxwell equations, evolution second order problems, variational solutions, asymptotic behavior

DOI: 10.3233/ASY-211730

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 289-320, 2022

Authors: Li, Chunyi | Song, Chaoqun | Quan, LiYan | Xiang, Jianhao | Xiang, Mingqi

Article Type: Research Article

Abstract: The aim of this paper is to consider the following fractional parabolic problem u t + ( − Δ ) p α u + ( − Δ ) q β u = f ( x , u ) ( x , t ) ∈ Ω × ( 0 , ∞ ) , u = 0 ( x , t ) ∈ ( R N ∖ Ω ) × ( 0 , ∞ ) , u ( …x , 0 ) = u 0 ( x ) x ∈ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) p α is the fractional p -Laplacian with 0 < α < 1 < p < ∞ , ( − Δ ) q β is the fractional q -Laplacian with 0 < β < α < 1 < q < p < ∞ , r > 1 and λ > 0 . The global existence of nonnegative solutions is obtained by combining the Galerkin approximations with the potential well theory. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions. Show more

Keywords: Fractional (p, q)-Laplacian, global existence, decay estimates

DOI: 10.3233/ASY-211731

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 321-338, 2022

Authors: Bazarra, Noelia | Fernández, José R. | Magaña, Antonio | Quintanilla, Ramón

Article Type: Research Article

Abstract: In this paper, we consider several problems arising in the theory of thermoelastic bodies with voids. Four particular cases are considered depending on the choice of the constitutive tensors, assuming different dissipation mechanisms determined by Moore–Gibson–Thompson-type viscosity. For all of them, the existence and uniqueness of solutions are shown by using semigroup arguments. The energy decay of the solutions is also analyzed for each case.

Keywords: Thermoviscoelasticity, existence and uniqueness, energy decay, dissipation mechanisms

DOI: 10.3233/ASY-211732

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 339-359, 2022

Authors: Chen, Zhi | Feng, Weixun | Qin, Dongdong

Article Type: Research Article

Abstract: In this paper we consider the following 2D MHD system with horizontal dissipation in a strip domain T × R . ∂ t u + u · ∇ u + ∂ 11 u + ∇ p = b · ∇ b , ∂ t b + u · ∇ b + ∂ 11 b = b · ∇ u , ∇ · u = ∇ · b = 0 , u ( 0 ) = …u 0 ( x ) , b ( 0 ) = b 0 ( x ) . A bootstrapping argument together with a more accurate energy functional is employed in order to get the stability for the above system. Moreover, using a suitable transform, we also investigate the 2D MHD system with vertical dissipation in a strip domain R × T . Show more

Keywords: 2D MHD system, horizontal dissipation, vertical dissipation, strip domain

DOI: 10.3233/ASY-211733

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 361-377, 2022

Authors: Serafin, G.

Article Type: Research Article

Abstract: We establish short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. So far, such results were only known in simple cases where explicit formulae are available, i.e., for sets as half-line, interval and their products. Presented asymptotics may be considered as a complement or a generalization of the famous “principle of not feeling the boundary” in case of a ball. Following the metaphor, the principle reveals when the process does not feel the boundary, while we describe what happens when it starts feeling the boundary.

Keywords: Heat kernel, ball, asymptotics, Laplacian, Brownian motion

DOI: 10.3233/ASY-211734

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 379-412, 2022

Authors: Emereuwa, Chigoziem | Mohammed, Mogtaba

Article Type: Research Article

Abstract: In this paper, we present new homogenization results of a stochastic model for flow of a single-phase fluid through a partially fissured porous medium. The model is a double-porosity model with two flow fields, one associated with the system of fissures and the other associated with the porous system. This model is mathematically described by a system of nonlinear stochastic partial differential equations defined on perforated domain. The main tools to derive the homogenized stochastic model are the Nguetseng’s two-scale convergence, tightness of constructed probability measures, Prokhorov and Skorokhod compactness process and Minty’s monotonicity method.

Keywords: Homogenization, single phase flow, partially fissured media, two-scale convergence, Minty’s monotonicity method, probabilistic compactness results, stochastic calculus

DOI: 10.3233/ASY-211735

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 413-450, 2022

Authors: Chen, Yuhui | Li, Minling | Yao, Qinghe | Yao, Zheng-an

Article Type: Research Article

Abstract: In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a …way to capture L 2 -convergence result via auxiliary logarithmic time decay estimates with the initial data in L p ( R 2 ) ∩ L 2 ( R 2 ) . Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized. Show more

Keywords: Viscoelastic fluids, sharp time decay rates, global-in-time stability, large solutions

DOI: 10.3233/ASY-211736

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 451-484, 2022

Authors: Zhang, Jing | Li, Lin

Article Type: Research Article

Abstract: In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , …where N ⩾ 4 , 0 ⩽ μ < μ ‾ , μ ‾ = ( N − 2 ) 2 4 , V is periodic in x , K and f are asymptotically periodic in x , we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1 ). Show more

Keywords: Schrödinger equation, generalized Nehari manifold, asymptotically periodic

DOI: 10.3233/ASY-211737

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 485-503, 2022

Authors: Naoyasu, Kita | Takuya, Sato

Article Type: Research Article

Abstract: This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u . Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The …other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 . Show more

Keywords: Nonlinear Schrödinger equation, decay estimate, critical dissipative nonlinearity

DOI: 10.3233/ASY-211738

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 505-517, 2022

Authors: Freitas, M.M. | Ramos, A.J.A. | Dos Santos, M.J. | Miranda, L.G.R. | Almeida, J.L.L.

Article Type: Research Article

Abstract: We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

Keywords: Mixture of solids, time delay, quasi-stability, global attractors, upper-semicontinuity

DOI: 10.3233/ASY-211739

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 519-544, 2022

Authors: dos Santos, Bruna C. | Oliva, Sergio M. | Rossi, Julio D.

Article Type: Research Article

Abstract: In this paper, we analyze a model composed by coupled local and nonlocal diffusion equations acting in different subdomains. We consider the limit case when one of the subdomains is thin in one direction (it is concentrated to a domain of smaller dimension) and as a limit problem we obtain coupling between local and nonlocal equations acting in domains of different dimension. We find existence and uniqueness of solutions and we prove several qualitative properties (like conservation of mass and convergence to the mean value of the initial condition as time goes to infinity).

Keywords: Nonlocal diffusion, heat equation, asymptotic behavior

DOI: 10.3233/ASY-211740

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 545-575, 2022

Authors: Boumaza, Nouri | Gheraibia, Billel

Article Type: Research Article

Abstract: In this paper, we consider the initial boundary value problem for the p -Laplacian equation with weak and p -Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases p > 2 and p = 2 . To our best knowledge, there is no results of the p -Laplacian equation with a nonlinear boundary delay term.

Keywords: p-Laplacian equation, strong damping, delay term, nonlinear boundary conditions, global existence, general decay, finite time blow up

DOI: 10.3233/ASY-211742

Citation: Asymptotic Analysis, vol. 129, no. 3-4, pp. 577-592, 2022