Asymptotic Analysis - Volume 103, issue 1-2

Authors: Khader, Maisa | Said-Houari, Belkacem

Article Type: Research Article

Abstract: We consider the Cauchy problem for the one-dimensional Bresse system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the solution using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. In fact we prove that the dissipation induced by the heat conduction is very weak and produces very slow decay rates. In addition in some cases, those decay rates are of regularity-loss type. Also, we prove that there is a number (depending on the parameters of the system) …that controls the decay rate of the solution and the regularity assumptions on the initial data. In addition, we show that in the absence of the frictional damping, the memory damping term is not strong enough to produce a decay rate for the solution. In fact, we show in this case, despite the fact that the energy is still dissipative, the solution does not decay at all. This result improves and extends several results, such as those in Appl. Math. Optim. (2016 ), to appear, Communications in Contemporary Mathematics 18 (4) (2016 ), 1550045, Math. Methods Appl. Sci. 38 (17) (2015 ), 3642–3652 and others. Show more

Keywords: The Bresse system, decay, regularity loss, heat conduction, Gurtin–Pipkin law, Lyapunov functional

DOI: 10.3233/ASY-171417

Citation: Asymptotic Analysis, vol. 103, no. 1-2, pp. 1-32, 2017

Authors: Ammari, Kaïs | Shel, Farhat | Vanninathan, Muthusamy

Article Type: Research Article

Abstract: In this paper we study the dynamic feedback stability for some simplified model of fluid–structure interaction on a tree. We prove that, under some conditions, the energy of the solutions of the dissipative system decay exponentially to zero when the time tends to infinity. Our technique is based on a frequency domain method and a special analysis for the resolvent.

Keywords: Dynamic feedback, asymptotic stabilization, resolvent method

DOI: 10.3233/ASY-171418

Citation: Asymptotic Analysis, vol. 103, no. 1-2, pp. 33-55, 2017

Authors: Miranville, Alain | Saoud, Wafa | Talhouk, Raafat

Article Type: Research Article

Abstract: Our aim in this article is to study the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system. In particular, we prove the existence of an exponential attractor and, as a consequence, the existence of the global attractor with finite fractal dimension.

Keywords: Allen–Cahn/Cahn–Hilliard system, global attractor, exponential attractor, fractal dimension

DOI: 10.3233/ASY-171419

Citation: Asymptotic Analysis, vol. 103, no. 1-2, pp. 57-76, 2017

Authors: Ikehata, Ryo

Article Type: Research Article

Abstract: We consider a mixed problem for wave equations with a localized damping in the n -dimensional half space R + n . We first propose a new type of Hardy inequality in the whole space by employing the Fourier transform. Then, this inequality combined with a multiplier method developed by [Sci. Math. Japon. 55 (2002 ), 33–42] will be effectively applied to the half space problems of the damped wave equations with a variable coefficient to catch a fast energy decay property. The Fourier transform is partially used to obtain the L …2 -boundedness of the solution. The method introduced in this paper will compensate lack of an effective Hardy type inequality in the n -dimensional half space. Show more

Keywords: Localized damping, wave equation, half space, weighted initial data, multiplier method, Fourier analysis, total energy decay

DOI: 10.3233/ASY-171420

Citation: Asymptotic Analysis, vol. 103, no. 1-2, pp. 77-94, 2017

Authors: Bresch, Didier | Renardy, Michael

Article Type: Research Article

Abstract: This short note concerns the formal limit presented in [J. Comput Phys 230 (2011), 8057–8088] and [Actes colloquia Rouen (2012)] between the compressible Euler equations with singular pressure (soft model) and the pressureless Euler system with unilateral constraint (hard model). These soft and hard models with maximal constraint on the density are used to reproduce congestion phenomena. In this paper, we are interested in the question how the different regions (congested and non-congested regions) fit together and how the transition occurs when congestion first develops in the initial system namely the compressible Euler equation with singular pressure. …We shall develop a formal solution by matched asymptotics and show that a shock front may separate the congested region from the outside non-congested region. This, in some sense, shows that to justify the formal limit between the two models (soft and hard Euler systems), the shock fronts formation has to be considered and mathematically analyzed. Show more

Keywords: Pressureless, congestion, shock fronts, singular point, boundary layers, compressible Euler

DOI: 10.3233/ASY-171421

Citation: Asymptotic Analysis, vol. 103, no. 1-2, pp. 95-101, 2017