On the lack of exponential stability for an elastic–viscoelastic waves interaction system

Zhang, Qiong

doi:10.1016/j.nonrwa.2017.02.019

Cited by 15 publications

(13 citation statements)

References 20 publications

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“…Indeed, if the material parameter b is smooth enough at the interface, then the energy of System (1) decays exponentially to zero as t goes to infinity under appropriate geometric conditions imposed on the damped region (see [19], [23], [28], [8]). However, Q. Zhang proved in [30] that the exponential decay fails in any geometry if the damping coefficient b is discontinuous along the interface. Note that, in this case, the lack of exponential stability still holds even in the 1 − d case (see [9]).…”

Section: 1mentioning

confidence: 99%

Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

Wehbe¹,

Nasser²,

Noun³

2021

MCRF

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<p style='text-indent:20px;'>We investigate a multidimensional transmission problem between viscoelastic system with localized Kelvin-Voigt damping and purely elastic system under different types of geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. It was shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system. So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. For this aim, using frequency domain approach combined with a new multiplier technic, we will establish a polynomial energy decay estimate of type <inline-formula><tex-math id="M1">\begin{document}$ t^{-1} $\end{document}</tex-math></inline-formula> for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type <inline-formula><tex-math id="M2">\begin{document}$ t^{-2/5} $\end{document}</tex-math></inline-formula> if the strip is localized near the boundary. Else, it's of type <inline-formula><tex-math id="M3">\begin{document}$ t^{-1/3} $\end{document}</tex-math></inline-formula>. The main novelty in this paper is that the geometric situations covered here are richer and less restrictive than those considered in [<xref ref-type="bibr" rid="b31">31</xref>], [<xref ref-type="bibr" rid="b28">28</xref>], [<xref ref-type="bibr" rid="b19">19</xref>] and include in particular an example where the damping region is localized faraway from the boundary. Note that part of the results of this paper was announced in [<xref ref-type="bibr" rid="b22">22</xref>].</p>

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Section: 1mentioning

confidence: 99%

Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

Wehbe¹,

Nasser²,

Noun³

2021

MCRF

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“…For the stability of higher dimensional wave equations with local Kelvin-Voigt damping, we refer the readers to the papers [2,7,8,19,26,27,32,34,35] and the reference therein.…”

Section: Introductionmentioning

confidence: 99%

Sharp stability of a string with local degenerate Kelvin–Voigt damping

2022

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This paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin–Voigt damping utt(x,t)badbreak−false[ux(x,t)+b(x)uxt(x,t)false]xgoodbreak=0,x∈(−1,1),tgoodbreak>0,\begin{equation} u_{tt}(x,t)-[u_{x}(x,t)+b(x) u_{xt}(x,t) ]_{x}=0,\; x\in (-1,1),\; t>0, \end{equation}where bfalse(xfalse)=0$b(x)=0$ on x∈false(−1,0false]$x\in (-1,0]$, and b(x)=xα>0$b(x)=x^\alpha >0$ on x∈false(0,1false)$x\in (0,1)$ for α∈false(0,1false)$\alpha \in (0,1)$. It is known that the optimal decay rate of solution is t−2$t^{-2}$ in the limit case α=0$\alpha =0$ and exponential for α≥1$\alpha \ge 1$. When α∈false(0,1false)$\alpha \in (0,1)$, the damping coefficient bfalse(xfalse)$b(x)$ is continuous, but its derivative has a singularity at the interface x=0$x=0$. In this case, the best known decay rate is t−3−α2(1−α)$t^{-\frac{3-\alpha }{2(1-\alpha )}}$, which fails to match the optimal one at α=0$\alpha =0$. In this paper, we obtain a sharper polynomial decay rate t−2−α1−α$t^{-\frac{2-\alpha }{1-\alpha }}$. More significantly, it is consistent with the optimal polynomial decay rate at α=0$\alpha =0$ and uniform boundedness of the resolvent operator on the imaginary axis at α=1$\alpha = 1$ (consequently, the exponential decay rate at α=1$\alpha =1$ as t→∞$t\rightarrow \infty$). This is a big step toward the goal of obtaining eventually the optimal decay rate.

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“…In [1], Liu and Rao proved the exponential stability of the system if damping coefficient function

D(x)\in C^{2}(\overline{\Omega }_{y})

\partial \Omega _{z}\subset \overline{\Omega }_{y}

, and

\overline{\Omega }=\overline{\Omega }_{y}\cup \overline{\Omega }_{z}

(see Figure 1). Zhang in [15] showed that if

D(x)

is discontinuous at the interface, the energy of the system always loses exponential stability for any geometry of the damping. Moreover, under certain geometric condition on the damping region, Zhang in [2] further studied the polynomial stability of the system and proved that the norm of its solution decays polynomially with

t^{-\frac{1}{2}}

with smooth initial states.…”

Section: Introductionmentioning

confidence: 99%

Longtime behavior of multidimensional wave equation with local Kelvin–Voigt damping

Han

Zhang

2022

Z Angew Math Mech

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In this paper, the longtime behavior of a coupled multidimensional elasticviscoelastic waves system is considered. This model consists of an elastic wave domain and an viscoelastic wave domain, connecting by a common interface.The dissipative damping is produced in the viscoelastic wave via the boundary connection. By the resolvent estimate together with microlocal analysis argument, we show that the corresponding semigroup is polynomially stable with decay rate 𝑡 −1 under certain conditions.

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On the lack of exponential stability for an elastic–viscoelastic waves interaction system

Cited by 15 publications

References 20 publications

Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions

Sharp stability of a string with local degenerate Kelvin–Voigt damping

Longtime behavior of multidimensional wave equation with local Kelvin–Voigt damping

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