A note on the energy transfer in coupled differential systems

Conti, Monica; Liverani, Lorenzo; Pata, Vittorino

doi:10.3934/cpaa.2021042

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…But even if the coupling is strong enough to bypass a certain critical threshold, the system may remain unstable if the action of the antidamping is more effective than the one of the damping. To the best of our knowledge, the first analysis of this kind has been made in the very recent paper [9], where a simple (yet not so simple) system of ODEs is considered, that is,…”

Section: A General Discussionmentioning

confidence: 99%

The MGT-Fourier model in the supercritical case

Conti¹,

Liverani²,

Pata³

2021

Preprint

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We address the energy transfer in the differential systemmade by a Moore-Gibson-Thompson equation in the supercritical regime, hence antidissipative, coupled with the classical heat equation. The asymptotic properties of the related solution semigroup depend on the strength of the coupling, ruling the competition between the Fourier damping and the MGT antidamping. Exponential stability will be shown always to occur, provided that the coupling constant is sufficiently large with respect to the other structural parameters. A fact of general interest will be also discussed, namely, the impossibility of attaining the optimal exponential decay rate of a given dissipative system via energy estimates.

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Section: A General Discussionmentioning

confidence: 99%

The MGT-Fourier model in the supercritical case

Conti¹,

Liverani²,

Pata³

2021

Preprint

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“…Although there is a vast literature concerning systems of coupled conservative-dissipative equations (e.g., [1,2,4,6,7,11,12,19,30], just to name a few), the behavior of systems like (2.1) remains largely unexplored. One paper in that direction is [9], dealing with the ODE model…”

Section: Introductionmentioning

confidence: 99%

Thermoelasticity with antidissipation

Conti¹,

Liverani²,

Pata³

2022

DCDS-S

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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<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} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ 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>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>.

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