Nonclassical dynamical thermoelasticity

Hetnarski, Richard B.; Ignaczak, Jòzef

doi:10.1016/s0020-7683(99)00089-x

Cited by 153 publications

(54 citation statements)

References 29 publications

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“…We note that µ * is the lowest eigenvalue of the matrix aλ −λ/2 −λ/2 ωa − 1 which is positive because of condition (9). From (13) and (15), we see that…”

Section: R Quintanilla Zampmentioning

confidence: 95%

“…In this sense, there are several parabolic and hyperbolic theories that describe the heat conduction, where the propagation of heat is modelled with finite propagation speed, in contrast to the classical model using Fourier's law leading to infinite propagation speed of heat signals. Good reviews are the articles by Chandrasekharaiah [1] or Hetnarski and Ignaczak [8,9]. Recently, Tzou [28][29][30][31] has proposed the dual-phase-lag heat equation.…”

Section: Introductionmentioning

confidence: 99%

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Spatial estimates for an equation with a delay term

Quintanilla¹

2009

Z. Angew. Math. Phys.

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In this note, we investigate the spatial behaviour of the solutions for a theory for the heat conduction with a delay term. We obtain an alternative of the Phragmen-Lindelof type. That is the solutions either decay in a exponential way or blow-up at infinity in a exponential way. We also describe how to obtain an upper bound for the amplitude term. It is worth noting that this is the first contribution on spatial behaviour for partial differential equations involving a delay term. We use the energy arguments to obtain our main results. The main point of the contribution is the use of a suitable weighted energy function.

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“…We note that µ * is the lowest eigenvalue of the matrix aλ −λ/2 −λ/2 ωa − 1 which is positive because of condition (9). From (13) and (15), we see that…”

Section: R Quintanilla Zampmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Spatial estimates for an equation with a delay term

Quintanilla¹

2009

Z. Angew. Math. Phys.

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“…From a physical point of view this is a drawback of the model because it predicts that heat waves propagate with infinite speed. To save the principle of causality, several heat conduction theories were suggested in the second part of the last century (see [2,6,7]). In the books [10,20,22], several studies concerning the applicability of nonclassical thermoelastic theories are considered.…”

Section: Introductionmentioning

confidence: 99%

Phase-lag heat conduction: decay rates for limit problems and well-posedness

Borgmeyer

Quintanilla²,

Racke

2014

J. Evol. Equ.

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In two recent papers the authors have studied conditions on the relaxation parameters in order to guarantee the stability or instability of solutions for the Taylor approximations to dual-phase-lag and three-phase-lag heat conduction equations. However, for several limit cases relating to the parameters the kind of stability was unclear. Here we analyze these limit cases and clarify whether we can expect exponential or slow decay for the solutions. Moreover, rather general well-posedness results for three-phase-lag models are presented. Finally, the exponential stability expected by spectral analysis is rigorously proved exemplarily.

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“…There are several parabolic and hyperbolic theories that describe the heat conduction, where the propagation of heat is modeled as having finite speed, in contrast to the classical model using Fourier's law and leading to infinite propagation speed of heat signals. The articles by Chandrasekharaiah [11] or Hetnarski and Ignaczak [12,13] and the books by Ignaczak and Ostoja-Starzewski [14] or Straughan [15] are good reviews of the situation.…”

Section: Introductionmentioning

confidence: 99%

Spatial behavior for solutions in heat conduction with two delays

Leseduarte

Quintanilla

2014

Applicable Analysis

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In this note we investigate the spatial behavior of the solutions of the equation proposed to describe a theory for the heat conduction with two delay terms. We obtain an alternative of the Phragmén-Lindelöf type, which means that the solutions either decay or blow-up at infinity, both options in an exponential way. We also describe how to obtain an upper bound for the amplitude term. This is the first contribution on spatial behavior for partial differential equations involving two delay terms. We use energy arguments. The main point of the contribution is the use of an exponentially weighted energy function.

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Nonclassical dynamical thermoelasticity

Cited by 153 publications

References 29 publications

Spatial estimates for an equation with a delay term

Spatial estimates for an equation with a delay term

Phase-lag heat conduction: decay rates for limit problems and well-posedness

Spatial behavior for solutions in heat conduction with two delays

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