Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III
Belkacem Said-Houari Radouane Rahali
Evolution Equations & Control Theory 2013, 2(2): 423-440 doi: 10.3934/eect.2013.2.423
In this paper, we investigate the decay property of a Timoshenko system in thermoelasticity of type III in the whole space where the heat conduction is given by the Green and Naghdi theory. Surprisingly, we show that the coupling of the Timoshenko system with the heat conduction of Green and Naghdi's theory slows down the decay of the solution. In fact we show that the $L^2$-norm of the solution decays like $(1+t)^{-1/8}$, while in the case of the coupling of the Timoshenko system with the Fourier or Cattaneo heat conduction, the decay rate is of the form $(1+t)^{-1/4}$ [25]. We point out that the decay rate of $(1+t)^{-1/8}$ has been obtained provided that the initial data are in $L^1( \mathbb{R})\cap H^s(\mathbb{R}), (s\geq 2)$. If the wave speeds of the first two equations are different, then the decay rate of the solution is of regularity-loss type, that is in this case the previous decay rate can be obtained only under an additional regularity assumption on the initial data. In addition, by restricting the initial data to be in $H^{s}\left( \mathbb{R}\right)\cap L^{1,\gamma }\left( \mathbb{R}\right) $ with $ \gamma \in \left[ 0,1\right] $, we can derive faster decay estimates with the decay rate improvement by a factor of $t^{-\gamma/4}$.
keywords: Timoshenko heat conduction Decay rate thermoelasticity.
The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system
Abdelaziz Soufyane Belkacem Said-Houari
Evolution Equations & Control Theory 2014, 3(4): 713-738 doi: 10.3934/eect.2014.3.713
In this paper, we consider the Bresse system with frictional damping terms. We investigated the relationship between the frictional damping terms, the wave speeds of propagation and their influence on the decay rate of the solution. We proved that in many cases the solution enjoys the decay property of regularity-loss type. We introduced a new assumption on the wave speeds that controls the behavior of the solution of the Bresse system. In addition, when the coefficient $l $ goes to zero, we showed that the solution of the Bresse system decays faster than the one of the Timoshenko system. This result seems to be the first one to give the decay rate of the solution of the Bresse system in unbounded domain.
keywords: wave speeds. Bresse system Decay rate Timoshenko system regularity loos
General decay estimates for a Cauchy viscoelastic wave problem
Belkacem Said-Houari Salim A. Messaoudi
Communications on Pure & Applied Analysis 2014, 13(4): 1541-1551 doi: 10.3934/cpaa.2014.13.1541
In this paper, we consider the Cauchy problem of a viscoelatic wave equation and by using the energy method in the Fourier space, we show general decay estimates of the solution. This result improves and generalizes some other results in the literature.
keywords: general estimate. Viscoelastic wave equation damping rate decay
Exponential decay for solutions to semilinear damped wave equation
Stéphane Gerbi Belkacem Said-Houari
Discrete & Continuous Dynamical Systems - S 2012, 5(3): 559-566 doi: 10.3934/dcdss.2012.5.559
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
keywords: stable set decay rate Strong damping positive initial energy. global existence
Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction
Belkacem Said-Houari Flávio A. Falcão Nascimento
Communications on Pure & Applied Analysis 2013, 12(1): 375-403 doi: 10.3934/cpaa.2013.12.375
The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
keywords: Viscoelastic wave equation boundary feedback relaxation function. source term
Long-time behavior of solutions of the generalized Korteweg--de Vries equation
Belkacem Said-Houari
Discrete & Continuous Dynamical Systems - B 2016, 21(1): 245-252 doi: 10.3934/dcdsb.2016.21.245
In this paper, we study the large-time behavior of solutions to the initial-value problem for the generalized Korteweg--de Vries equation. We show that for initial data in some weighted space, the asymptotic behavior of the solution can be improved. In addition, we give the asymptotic profile of the fundamental solution of the linearized model. We extend and improve the results in [3] and [2].
keywords: rate damping global existence. decay KdV equation
Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks
Mokhtar Kirane Belkacem Said-Houari Mohamed Naim Anwar
Communications on Pure & Applied Analysis 2011, 10(2): 667-686 doi: 10.3934/cpaa.2011.10.667
We study the exponential stability of the Timoshenko beam system by interior time-dependent delay term feedbacks. The beam is clamped at the two hand points subject to two internal feedbacks: one with a time-varying delay and the other without delay. Using the variable norm technique of Kato, it is proved that the system is well-posed whenever an hypothesis between the weight of the delay term in the feedback, the weight of the term without delay and the wave speeds. By introducing an appropriate Lyapunov functional the exponential stability of the system is proved. Under the imposed constrain on the weights of the feedbacks and the wave speeds, the exponential decay of the energy is established via a suitable Lyapunov functional.
keywords: global solutions damping delay exponential decay. stability Timoshenko

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