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### Open Access Journals

EECT

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 $t^{-\gamma/4}$.

EECT

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.

CPAA

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.

DCDS-S

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].

CPAA

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.

DCDS-B

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].

CPAA

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.

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