Asymptotic Analysis - Volume 132, issue 1-2

Authors: Raimondi, Federica

Article Type: Research Article

Abstract: This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε -periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ε γ . We prove an homogenization result for − 1 < γ < 1 by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 …(2008 ) 1585–1620 and The Periodic Unfolding Method (2018 ) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011 ) 891–927). One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence { u ε } of the solutions, as ε → 0 . More precisely, our result shows that the gradient of u ε behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate. Show more

Keywords: Two-component domains, Homogenization, Periodic Unfolding Method, Quasilinear elliptic equations, Singular equations

DOI: 10.3233/ASY-221783

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 1-27, 2023

Authors: Feng, Baowei | Messaoudi, Salim | Soufyane, Abdelaziz | Zahri, Mostafa

Article Type: Research Article

Abstract: In this paper, we study the stability of a Bresse system with memory-type boundary conditions. For a wider class of kernel functions, we establish an optimal explicit energy decay result. Our stability result improves many earlier results in the literature. Finally, we also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme.

Keywords: Bresse system, Timoshenko, viscoelastic damping, general decay, convexity

DOI: 10.3233/ASY-221784

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 29-60, 2023

Authors: Zhang, Ning | Tang, Xianhua | Chen, Sitong

Article Type: Research Article

Abstract: In this paper, we prove the existence of nontrivial solutions for the following planar quasilinear Schrödinger equation: − Δ u + V ( x ) u − Δ ( u 2 ) u = g ( u ) , x ∈ R 2 , where V ∈ C ( R 2 , [ 0 , ∞ ) ) and g ∈ C ( R , R ) is of subcritical exponential growth satisfying some …mild conditions. In particular, by means of the Trudinger–Moser inequality, we give a different method from the one of the polynomial growth nonlinearities to prove the Brézis–Lieb split property when f has subcritical exponential growth. Our result extends and complements the one of Chen–Rădulescu–Tang–Zhang (Rev. Mat. Iberoam. 36 (2020 ) 1549–1570) dealing with the higher dimensions N ⩾ 3 to the dimension N = 2 . Show more

Keywords: Quasilinear Schrödinger equation, nontrivial solution, exponential growth

DOI: 10.3233/ASY-221785

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 61-82, 2023

Authors: Korotyaev, Evgeny | Mokeev, Dmitrii

Article Type: Research Article

Abstract: We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems: in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of zeros of the reflection coefficient for another compactly supported potential, 2) the set of “isoresonance potentials” is described, 3) the forbidden domain for resonances is estimated, 4) asymptotics of the resonances counting function is determined, 5) these …results are applied to canonical systems. Show more

Keywords: Dirac operators, inverse problems, resonances, canonical systems, compactly supported potentials

DOI: 10.3233/ASY-221786

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 83-130, 2023

Authors: Khalili, Zineb | Ouchenane, Djamel

Article Type: Research Article

Abstract: The main goal of this paper is to investigate the exponential stability of the Timoshenko system in thermoelasticity of second sound with a time-varying delay term in the internal feedback. The well-posedness of the problem is assured by using the variable norm technique of Kato. Furthermore the stability of the system is shown by applying the energy method.

Keywords: Timoshenko system, thermoelasticity, second sound, linear damping, exponential decay

DOI: 10.3233/ASY-221787

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 131-152, 2023

Authors: Freitas, M.M. | Santos, M.L. | Dos Santos, M.J. | Ramos, A.J.A.

Article Type: Research Article

Abstract: In this paper we investigate the long-term behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation and nonlinear feedback force. We prove that the porous-elasticity problem converges to a quasi-static problem for the microvoids motion as a suitable parameter J tends to zero. Finite dimensional global attractor with additional regularity in J is obtained using the recent quasi-stability theory. Finally, we compare the porous-elasticity problem with quasi-static problem, in the sense of the upper-semicontinuity of their attractors as J → 0 .

Keywords: Porous-elasticity, quasi-static microvoids, singular limit, global attractor, upper-semicontinuity

DOI: 10.3233/ASY-221788

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 153-174, 2023

Authors: Goudey, Rémi

Article Type: Research Article

Abstract: We consider an homogenization problem for the second order elliptic equation − div ( a ( · / ε ) ∇ u ε ) = f when the coefficient a is almost translation-invariant at infinity and models a geometry close to a periodic geometry. This geometry is characterized by a particular discrete gradient of the coefficient a that belongs to a Lebesgue space L p ( R d ) for p ∈ [ 1 , + ∞ [ . When p …< d , we establish a discrete adaptation of the Gagliardo–Nirenberg–Sobolev inequality in order to show that the coefficient a actually belongs to a certain class of periodic coefficients perturbed by a local defect. We next prove the existence of a corrector and we identify the homogenized limit of u ε . When p ⩾ d , we exhibit admissible coefficients a such that u ε possesses different subsequences that converge to different limits in L 2 . Show more

Keywords: Homogenization, elliptic PDEs, corrector equation

DOI: 10.3233/ASY-221789

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 175-216, 2023

Authors: Hillairet, M. | Sabbagh, L.

Article Type: Research Article

Abstract: We consider the motion of spherical particles in the whole space R 3 filled with a viscous fluid. We show that, when modelling the fluid behavior with an incompressible Stokes system, solutions are global and no collision occurs between the spheres in finite time.

Keywords: Fluid-solid systems, Stokes equations, contact issue, global-in-time existence of solutions

DOI: 10.3233/ASY-221790

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 217-243, 2023

Authors: El Ouaarabi, Mohamed | Allalou, Chakir | Melliani, Said

Article Type: Research Article

Abstract: In this article, we consider a Neumann boundary value problem driven by p ( x ) -Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: − Δ p ( x ) l u + δ | u | ζ ( x ) − 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) in  Ω , ∂ …u ∂ η = 0 on  ∂ Ω , where Δ p ( x ) l u is the p ( x ) -Laplacian-like operator, Ω is a smooth bounded domain in R N , δ , μ and λ are three real parameters, p ( x ) , ζ ( x ) ∈ C + ( Ω ‾ ) , η is the outer unit normal to ∂ Ω and g , f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem. Show more

Keywords: Neumann boundary value problem, p(x)-Laplacian-like operator, capillarity phenomena, weak solution, topological degree methods, variable exponent Sobolev spaces

DOI: 10.3233/ASY-221791

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 245-259, 2023

Authors: Dhif, R. | Meftahi, H. | Rjaibi, B.

Article Type: Research Article

Abstract: In this paper, we consider the geometric inverse problem of recovering an obstacle ω immersed in a bounded fluid flow Ω governed by the time-dependent Brinkman model. We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on …the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method. Show more

Keywords: Optimal shape design, inverse problem, time-dependent Brinkman equation, topological sensitivity analysis

DOI: 10.3233/ASY-221792

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 261-284, 2023

Authors: Jiang, Yan | Liu, Hongyu | Zhang, Jiachuan | Zhang, Kai

Article Type: Research Article

Abstract: Consider the transmission eigenvalue problem for u ∈ H 1 ( Ω ) and v ∈ H 1 ( Ω ) : ∇ · ( σ ∇ u ) + k 2 n 2 u = 0 in  Ω , Δ v + k 2 v = 0 in  Ω , u = v , σ ∂ u ∂ ν = ∂ v ∂ ν …on  ∂ Ω , where Ω is a ball in R N , N = 2 , 3 . If σ and n are both radially symmetric, namely they are functions of the radial parameter r only, we show that there exists a sequence of transmission eigenfunctions { u m , v m } m ∈ N associated with k m → + ∞ as m → + ∞ such that the L 2 -energies of v m ’s are concentrated around ∂ Ω . If σ and n are both constant, we show the existence of transmission eigenfunctions { u j , v j } j ∈ N such that both u j and v j are localized around ∂ Ω . Our results extend the recent studies in (SIAM J. Imaging Sci. 14 (2021 ), 946–975; Chow et al.). Through numerics, we also discuss the effects of the medium parameters, namely σ and n , on the geometric patterns of the transmission eigenfunctions. Show more

Keywords: Transmission eigenfunctions, spectral geometry, boundary localization, wave localization

DOI: 10.3233/ASY-221794

Citation: Asymptotic Analysis, vol. 132, no. 1-2, pp. 285-303, 2023