Searcharxiv⌕ Search

arXiv subjects

Fathi Hassine

Publications and source records attributed to Fathi Hassine.

18 recordsLinked to original sources
arXiv2024

Stability of a degenerate thermoelastic equation

This work is dedicated to the study of a linear model arising in thermoelastic rod of homogeneous material. The system is resulting from a coupling of a heat and a wave equation in the interval $(0,1)$ with Dirichlet boundary conditions at the outer endpoints where the parabolic component is degenerating at the end point $x=0$. Two models are considered the first is with weak degeneracy and the second is with strong degeneracy. We aim to study the well-posedness and asymptotic stability of both systems using techniques from the $C_{0}$-semigroup theory and a use a frequency domain approach based on the well-known result of Pr\"uss in order to prove using some multiplier techniques that the energy of classical solutions decays uniformly as time goes to infinity.

math.AP↗
arXiv2023

Uniform stabilization of an acoustic system

We study the problem of stabilization for the acoustic system with a spatially distributed damping. With imposing hypothesis on the structural properties of the damping term, we identify exponential decay of solutions with growing time.

math.AP↗
arXiv2021

Stabilization for Euler-Bernoulli beam equation with a local degenerated Kelvin-Voigt damping

We consider the Euler-Bernoulli beam equation with a local Kelvin-Voigt dissipation type in the interval $(-1,1)$. The coefficient damping is only effective in $(0,1)$ and is degenerating near the $0$ point with a speed at least equal to $x^α$ where $α\in(0,5)$. We prove that the semigroup corresponding to the system is polynomially stable and the decay rate depends on the degeneracy speed $α$.

math.AP↗
arXiv2020

Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface

In this paper, we study the stability problem of a star-shaped network of elastic strings with a local Kelvin-Voigt damping. Under the assumption that the damping coefficients have some singularities near the transmission point, we prove that the semigroup corresponding to the system is polynomially stable and the decay rates depends on the speed of the degeneracy. This result improves the decay rate of the semigroup associated to the system on an earlier result of Z.~Liu and Q.~Zhang in \cite{LZ} involving the wave equation with local Kelvin-Voigt damping and non-smooth coefficient at interface.

math.AP↗
arXiv2020

Logarithmic stabilization of an acoustic system with a damping term of Brinkman type

We study the problem of stabilization for the acoustic system with a spatially distributed damping. Without imposing any hypotheses on the structural properties of the damping term, we identify logarithmic decay of solutions with growing time. Logarithmic decay rate is shown by using a frequency domain method and combines a contradiction argument with the multiplier technique and a new Carleman estimate to carry out a special analysis for the resolvent.

math.AP↗
arXiv2019

Stability for coupled waves with locally disturbed Kelvin-Voigt damping

We consider a coupled wave system with partial Kelvin-Voigt damping in the interval (-1,1), where one wave is dissipative and the other does not. When the damping is effective in the whole domain (-1,1) it was proven in H.Portillo Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt damping, Applied Mathematics Letters 67 (2017), 16-20. That the energy is decreasing over the time with a rate equal to $t^{-\frac{1}{2}}$. In this paper, using the frequency domain method we show the effect of the coupling and the non smoothness of the damping coefficient on the energy decay. Actually, as expected we show the lack of exponential stability, that the semigroup loses speed and it decays polynomially with a slower rate then given in, H.Portillo Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt damping, Applied Mathematics Letters 67 (2017), 16-20, down to zero at least as $t^{-\frac{1}{12}}$.

math.AP↗
arXiv2019

Fractional-feedback stabilization for a class of evolution systems

We study the problem of stabilization for a class of evolution systems with fractional-damping. After writing the equations as an augmented system we prove in this article first that the problem is well posed. Second, using the LaSalle's invariance principle we show that the energy of the system is Strongly stable. Then, based on a resolvent approach we show a luck of uniform stabilization. Next, using multiplier techniques combined with the frequency domain method, we shall give a polynomially stabilization result under some consideration on the stabilization of an auxiliary dissipating system. Finally, we give some applications to the wave equation.

math.AP↗
arXiv2019

Stabilization for vibrating plate with singular structural damping

We consider the dynamic elasticity equation, modeled by the Euler-Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic ) damping in a boundary domain. Using a frequency domain method combined, based on the Burq's result, combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity.

math.AP↗
arXiv2019

Stabilization of fractional-evolution systems

This paper is devoted to the analysis of the problem of stabilization of fractional (in time) partial differential equations. We consider the following equation $$ \partial^{α,η}_{t} u(t)=\mathcal{A}u(t)-\fracη{Γ(1-α)}\int_{0}^{t}(t-s)^{-α} \, e^{-η(t-s)}u(s)\, ds,\; t > 0, $$ with the initial data $u(0)=u^{0}$, where $\mathcal{A}$ is a unbounded operator in Hilbert space and $\partial_{t}^{α,η}$ stands for the fractional derivative. We provide two main results concerning the behavior of the solutions when $t\longrightarrow+\infty$. We look first to the case $η>0$ where we prove that the solution of this problem is exponential stable then we consider the case $η=0$ when we prove under some consideration on the resolvent that the energy of the solution goes to $0$ as $t$ goes to the infinity as $1/t^α$.

math.AP↗
arXiv2018

Remark on the pointwise stabilization of an elastic string equation

We consider an initial and boundary value problem the one dimensional wave equation with damping concentrated at an interior point. We prove a result of a logarithmic decay of the energy of a system with homogeneous Dirichlet boundary conditions. The method used is based on the resolvent estimate approach which derives from the Carleman estimate technique. Under an algebraic assumption describing the right location of the actuator, we prove a logarithmic decay of the energy of solution. We show that this assumption is lower than the one given by [Tuc] and [AHT] which depends on the diophantine approximations properties of the actuator's location.

math.OC↗
arXiv2018

Stability of elastic transmission systems with a local Kelvin-Voigt damping

In this paper, we consider the longitudinal and transversal vibrations of the transmission Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam and only in one side of the transmission point. We prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is polynomially stable. Due to the locally distributed and unbounded nature of the damping, we use a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

math.AP↗
arXiv2018

Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping

We consider a beam and a wave equations coupled on an elastic beam through transmission conditions. The damping which is locally distributed acts through one of the two equations only; its effect is transmitted to the other equation through the coupling. First we consider the case where the dissipation acts through the beam equation. Using a recent result of Borichev and Tomilov on polynomial decay characterization of bounded semigroups we provide a precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this system is also polynomially stable and we provide precise polynomial decay estimates for its energy. Finally, we show the lack of uniform exponential decay of solutions for both models.

math.OC↗
arXiv2018

Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping

Let a fourth and a second order evolution equations be coupled via the interface by transmission conditions, and suppose that the first one is stabilized by a localized distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behavior of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay logarithmically at infinity even the feedback dissipation affects an arbitrarily small open subset of the interior. The method used, in this case, is based on a frequency method, and this by combining a contradiction argument with the Carleman estimates technique to carry out a special analysis for the resolvent.

math.AP↗
arXiv2018

Stabilization for the wave equation with singular Kelvin-Voigt damping

We consider the wave equation with Kelvin-Voigt damping in a bounded domain. The exponential stability result proposed by Liu and Rao or Tébou for that system assumes that the damping is localized in a neighborhood of the whole or a part of the boundary under some consideration. In this paper we propose to deal with this geometrical condition by considering a singular Kelvin-Voigt damping which is localized faraway from the boundary. In this particular case it was proved by Liu and Liu the lack of the uniform decay of the energy. However, we show that the energy of the wave equation decreases logarithmically to zero as time goes to infinity. Our method is based on the frequency domain method. The main feature of our contribution is to write the resolvent problem as a transmission system to which we apply a specific Carleman estimate.

math.AP↗
arXiv2015

A review of depth-averaged models for dry granular flows: Savage-Hutter and $μ(I)$-rheology

In this work we present a review of the most popular depth-averaged models to simulate dry granular flows such as aerial avalanches. The classical Savage-Hutter model and recent ones using a $μ(I)$-rheology law are studied. The objective is firstly to point out the advantages of each such models and secondly to understand how the hypothesis considered in the derivation process influence on the final system.

math.AP↗
arXiv2014

Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping

In this paper we will study the asymptotic behaviour of the energy decay of a transmission plate equation with locally distributed Kelvin-Voigt feedback. Precisly, we shall prove that the energy decay at least logarithmically over the time. The originality of this method comes from the fact that using a Carleman estimate for a transmission second order system which will be derived from the plate equation to establish a resolvent estimate which provide, by the famous Burq's result [Bur98], the kind of decay mentionned above.

math.AP↗
arXiv2013

Weak stabilization of a transmission Euler-Bernoulli plate equation with force and moment feedback

In this paper we will study the asymptotic behaviour of the energy decay of a transmission plate equation with force and moment feedback. Precisly, we shall prove that the energy decay at least logarithmically over the time. The method consist to use the classical second order Carleman estimate to estabish a resolvent estimate which provide by the famous Burq's result [Bur98] the kind of decay above mentionned.

math.AP↗