Logarithmic Decay of Wave Equation with Kelvin-Voigt Damping

Robbiano, Luc; Zhang, Qiong

doi:10.3390/math8050715

Cited by 9 publications

(3 citation statements)

References 26 publications

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“…However, the author still requires an inequality constraint on the gradient of the damping coefficient, which will not be required in our current study. In [31] the authors consider the equation y tt −div[∇y(t, x)+a(x)∇y t (t, x)] = 0 in a domain Ω ⊂ R d , and a(x) ∈ L 1 (Ω) and they show the logarithmic decay rate for energy of the system without any geometric assumption on the subdomain on which the damping is effective. There are two main difficulties regarding problem (1.1).…”

Section: Remarkmentioning

confidence: 99%

Uniform stabilization of the Klein-Gordon system

Cavalcanti¹,

Delatorre²,

Soares³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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We consider the Klein-Gordon system posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [5]. Although the present problem has some similarity to the reference [6] it is important to mention that due to the Kelvin-Voigt dissipation character associated with the nonlinearity of the problem the approach used is completely new, which is the main purpose of this paper.

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Section: Remarkmentioning

confidence: 99%

Uniform stabilization of the Klein-Gordon system

Cavalcanti¹,

Delatorre²,

Soares³

et al. 2020

Communications on Pure &Amp; Applied Analysis

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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

Self Cite

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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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“…Also, they proved a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and they applied it on a square domain where the damping is localized in a vertical strip. Also, in [34], the authors analyzed the long time behavior of the wave equation with local Kelvin-Voigt damping where they showed the logarithmic decay rate for energy of the system without any geometric assumption on the subdomain on which the damping is effective. Furthermore, in [10], the author showed how perturbative approaches and the black box strategy allow to obtain decay rates for Kelvin-Voigt damped wave equations from quite standard resolvent estimates (for example Carleman estimates or geometric control estimates).…”

Section: Introductionmentioning

confidence: 99%

A N-dimensional elastic\viscoelastic transmission problem with Kelvin-Voigt damping and non smooth coefficient at the interface

Akil¹,

Issa²,

Wehbe³

2021

Preprint

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We investigate the stabilization of a multidimensional system of coupled wave equations with only one Kelvin-Voigt damping. Using a unique continuation result based on a Carleman estimate and a general criteria of Arendt-Batty, we prove the strong stability of the system in the absence of the compactness of the resolvent without any geometric condition. Then, using a spectral analysis, we prove the non uniform stability of the system. Further, using frequency domain approach combined with a multiplier technique, we establish some polynomial stability results by considering different geometric conditions on the coupling and damping domains. In addition, we establish two polynomial energy decay rates of the system on a square domain where the damping and the coupling are localized in a vertical strip.

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Logarithmic Decay of Wave Equation with Kelvin-Voigt Damping

Cited by 9 publications

References 26 publications

Uniform stabilization of the Klein-Gordon system

Uniform stabilization of the Klein-Gordon system

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

A N-dimensional elastic\viscoelastic transmission problem with Kelvin-Voigt damping and non smooth coefficient at the interface

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