Optimal energy decay in a one-dimensional wave-heat system with infinite heat part

S., Ng, Abraham C.; Seifert, David

doi:10.1016/j.jmaa.2019.123563

Cited by 9 publications

(2 citation statements)

References 17 publications

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“…Similar one-dimensional systems with damping have been previously studied using the asymptotic theory of operator semigroups (see Refs. [1,5,15,20,23,24,28]).…”

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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“…Similar one-dimensional systems with damping have been previously studied using the asymptotic theory of operator semigroups (see Refs. [1,5,15,20,23,24,28]).…”

Section: Introductionmentioning

confidence: 99%

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

2022

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“…Their intrinsic mathematical interest apart, the main motivation for studying such systems stems from the fact that they can been viewed as linearisations of more complex fluid-structure models arising in fluid mechanics; see for instance [2,25,33]. In the absence of the integral term, (1.1) reduces to the classical waveheat system, whose asymptotic properties have been extensively analysed in the literature; see for instance [1,3,5,15,23,32,33] and the references therein. In particular, it is known that in this case the associated solution semigroup is semi-uniformly stable in the sense that all classical solutions converge to zero at a uniform rate, and more specifically the semigroup is polynomially stable with optimal decay rate t −2 as t → ∞.…”

Section: Introductionmentioning

confidence: 99%

Optimal decay for a wave-heat system with Coleman-Gurtin thermal law

Dell’Oro¹,

Paunonen²,

Seifert³

2021

Preprint

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We study the asymptotic behaviour of solutions to a one-dimensional coupled wave-heat system with Coleman-Gurtin thermal law. As our main results, we represent the system as a feedback interconnection between the wave part and the Coleman-Gurtin part and, using the asymptotic theory of C 0 -semigroups, we show that the associated semigroup in the history framework of Dafermos is polynomially stable with optimal decay rate t −2 as t → ∞. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.

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The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect

Akil,

Fragnelli,

Issa

2024

Mathematische Nachrichten

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In this paper, we investigate the stabilization of the transmission problem of the degenerate wave equation and the heat equation under the Coleman–Gurtin heat conduction law or Gurtin–Pipkin law with the memory effect. We investigate the polynomial stability of this system when employing the Coleman–Gurtin heat conduction, establishing a decay rate of type . Next, we demonstrate exponential stability in the case when Gurtin–Pipkin heat conduction is applied.

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Optimal energy decay in a one-dimensional wave-heat system with infinite heat part

Cited by 9 publications

References 17 publications

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

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

Optimal decay for a wave-heat system with Coleman-Gurtin thermal law

The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect

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