General boundary stabilization of memory type in thermoelasticity of type III

Messaoudi, Salim A.; Al-Shehri, Aisha

doi:10.1007/s00033-010-0102-3

Cited by 6 publications

(2 citation statements)

References 23 publications

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“…Finally, f (u), g(v) are nonlinear source terms. This problem has been studied in [27] with second sound and in [28][29][30][31][32] additionally with a viscoelastic dissipation acting on a part of the boundary, but without source term in the parabolic equation g(θ) = 0, without damping δ = 0 and without the viscoelastic term in the parabolic equation. An abstract formulation is studied in [33] with Cattaneo's law and inertial terms.…”

Section: Thermoelasticity System In N Dimensions With Short Memorymentioning

confidence: 99%

Global Non-existence of a Coupled Parabolic-Hyperbolic System of Thermoelastic Type with History

Esquivel-Avila

2023

Preprint

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We consider two abstract systems of parabolic-hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that in our framework are produced by the blow-up in finite time of the norm of the solution in the phase space. We employ a differential inequality to find sufficient conditions that produce the blow-up. To that end, we construct a set that is positive invariant for any positive value of the initial energy. As a result we found that the coupling with the parabolic equation stabilizes the system, as well as the damping term in the hyperbolic equation. Moreover, for any pair of positive values $(\xi,\epsilon)$, there exist initial data such that the corresponding solution with initial energy $\xi$ blows-up at a finite time less than $\epsilon$. Our purpose is to improve results previously published in the literature.

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Section: Thermoelasticity System In N Dimensions With Short Memorymentioning

confidence: 99%

Global Non-existence of a Coupled Parabolic-Hyperbolic System of Thermoelastic Type with History

Esquivel-Avila

2023

Preprint

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“…Finally, f (u), g(v) are nonlinear source terms. This problem was studied in [23] with second sound, and in [24,[29][30][31][32], it was investigated with viscoelastic dissipation acting on a part of the boundary, but without the source term in the parabolic equation g(θ) = 0, without damping δ = 0, and without the viscoelastic term in the parabolic equation. An abstract formulation was studied in [33] with Cattaneo's law and inertial terms.…”

Section: Thermoelastic System In N-dimensions With a Short Memorymentioning

confidence: 99%

Global Non-Existence of a Coupled Parabolic–Hyperbolic System of Thermoelastic Type with History

Esquivel-Avila

2023

Mathematics

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We consider two abstract systems of parabolic–hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that, in our framework, are produced by the blow-up in finite time of the norm of the solution in the phase space. We employ a differential inequality to find sufficient conditions that produce the blow-up. To that end, we construct a set that is positive-invariant for any positive value of the initial energy. As a result, we found that the coupling with the parabolic equation stabilizes the system, as well as the damping term in the hyperbolic equation. Moreover, for any pair of positive values (ξ,ϵ), there exist initial data, such that the corresponding solution with initial energy ξ blows up at a finite time less than ϵ. Our purpose is to improve results previously published in the literature.

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