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Discrete & Continuous Dynamical Systems - S
Article . 2022 . Peer-reviewed
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zbMATH Open
Article . 2022
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Thermoelasticity with antidissipation

Authors: Conti, M; Liverani, L; Pata, V;

Thermoelasticity with antidissipation

Abstract

<p style='text-indent:20px;'>We provide a complete stability analysis for the abstract differential system made by an antidamped wave-type equation, coupled with a dissipative heat-type equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} u_{tt} + A u -\gamma u_t = p A^{\alpha} \theta \\ \theta_{t} + \kappa A^{\beta} \theta = - p A^{\alpha} u_t \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ A $\end{document}</tex-math></inline-formula> is a strictly positive selfadjoint operator on a Hilbert space, <inline-formula><tex-math id="M2">\begin{document}$ \gamma, \kappa&gt;0 $\end{document}</tex-math></inline-formula>, and both the parameters <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> can vary between <inline-formula><tex-math id="M5">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>. The asymptotic properties of the associated solution semigroup are determined by the strength of the coupling, as well as the quantitative balance between the antidamping <inline-formula><tex-math id="M7">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> and the damping <inline-formula><tex-math id="M8">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula>. Depending on the value of <inline-formula><tex-math id="M9">\begin{document}$ (\alpha, \beta) $\end{document}</tex-math></inline-formula> in the unit square, one of the following mutually disjoint situations can occur: either the related semigroup decays exponentially fast, or all the solutions vanish but not uniformly, or there exists a trajectory whose norm blows up exponentially fast as <inline-formula><tex-math id="M10">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'> </p><p style='text-indent:20px;'>Correction: Sections 7, 8 and 9 are missing from this article. Such sections were present and peer-reviewed in the original submission, but they were mistakenly omitted during the preparation of the final version with the AIMS template. They are added in <a href="https://www.aimsciences.org/article/doi/10.3934/dcdss.2022125" target="_blank">Correction to “Thermoelasticity with antidissipation” (volume 15, number 8, 2022, 2173-2188)</a>.</p>

Country
Italy
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Keywords

damping and anti-damping, exponential blow up, PDEs in connection with classical thermodynamics and heat transfer, exponential stability, Asymptotic behavior of solutions to PDEs, damping and antidamping, stability, Fourier heat conduction law, Thermal effects in solid mechanics, thermoelasticity, Classical linear elasticity, Thermoelasticity

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
3
Top 10%
Top 10%
Average
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