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Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 2
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Original Articles

Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law

Pages 424-440 | Received 29 May 2011, Accepted 15 Sep 2011, Published online: 10 Oct 2011
 

Abstract

In this article, we investigate the decay properties of the linear thermoelastic plate equations in the whole space for both Fourier and Cattaneo's laws of heat conduction. We point out that while the paradox of infinite propagation speed inherent in Fourier's law is removed by changing to the Cattaneo law, the latter always leads to a loss of regularity of the solution. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We prove the decay estimates for initial data U 0 ∈ H s (ℝ) ∩ L 1(ℝ). In addition, by restricting the initial data to U 0 ∈ H s (ℝ) ∩ L 1,γ(ℝ) and γ ∈ [0, 1], we can derive faster decay estimates with the decay rate improvement by a factor of t −γ/2.

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Acknowledgement

This work has been supported by KAUST.

Notes

Notes

1. The Fourier law was the first constitutive relation of heat flux and was proposed by the French mathematical physicist Joseph Fourier in 1807.

2. The constant τ0 represents the time lag needed to establish the steady state of the heat conduction in an element of volume when a temperature gradient is suddenly imposed on that element. For most solid materials, τ0 varies from 10−10 s to 10−14 s. For gases, τ0 is in the range of 10−8 s up to 10−10 s. See the survey paper Citation36 for more details.

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