The MGT-Fourier model in the supercritical case

Conti, Monica; Liverani, Lorenzo; Pata, Vittorino

doi:10.1016/j.jde.2021.08.030

Cited by 12 publications

(9 citation statements)

References 33 publications

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“…In system (2.2), the two equations are coupled by velocities, but in fact the picture does not substantially change with a different coupling. The same phenomenon occurs for more complex PDE systems: for instance, [10] considers…”

Section: Introductionmentioning

confidence: 69%

“…Precisely, the uniform decay of the energy occurs if and only if κ > 0, whereas the equation exhibits an antidissipative behavior when κ < 0 (see [20]). In [10], system (2.3) is studied in the latter regime, so that there is a competition between the antidissipative mechanism of the MGT equation and the damping effect provided by the Fourier law. Once again, the authors showed that the exponential decay of the energy can be obtained only if the coupling parameter p is large enough in modulus.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

Thermoelasticity with antidissipation

Conti¹,

Liverani²,

Pata³

2022

DCDS-S

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“…The only difference is that now we do not know that S(t) is bounded. However, it has been proved in [6] that S(t) is exponentially stable if = 1 . In this situation, S(t) is certainly bounded and, from the general theory of C 0 -semigroups (see for instance [9,19]), we also know that Proposition 4.1 holds true.…”

Section: The Critical Casementioning

confidence: 99%

“…As shown in [1], in the subcritical case the semigroup is exponentially stable, namely, there exist 𝜔 > 0 and M ≥ 1 such that where ‖ ⋅ ‖ denotes the operator norm. Moreover, it has been recently proved in [6] that there exists a structural threshold independent of , of the form such that exponential stability occurs whenever…”

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confidence: 99%

On the analyticity of the abstract MGT-Fourier system

Dell’Oro

Pata

2022

Meccanica

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We consider the abstract MGT-Fourier system $${\left\{ \begin{array}{ll} u_{ttt}+\alpha u_{tt}+\beta A^\varrho u_t+\gamma A^\varrho u = \eta A \theta \\ \theta _t+\kappa A \theta = -\eta A u_{tt} - \eta \alpha A u_t \end{array}\right. }$$ u ttt + α u tt + β A ϱ u t + γ A ϱ u = η A θ θ t + κ A θ = - η A u tt - η α A u t in the subcritical regime $$\beta >\gamma /\alpha$$ β > γ / α , where the operator A is strictly positive selfadjoint. For any fixed $$\varrho \in [1,2]$$ ϱ ∈ [ 1 , 2 ] , we show that the associated solution semigroup S(t) is analytic and exponentially stable as well.

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“…the latter norm being the operator norm on the Banach space L(H) of bounded linear operators on H. When a semigroup is exponentially stable, it is of great importance to find the optimal decay rate, that is, the supremum of the ω > 0 for which (1) holds for some M = M (ω). And a related question is to establish whether or not such a supremum is attained, as the constant M might blow up as ω approaches the optimal decay rate.…”

mentioning

confidence: 99%