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Porous-elastic Plates: Fourier Versus Type III

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Abstract

In this paper we investigate the time decay of the solutions for a thermoelastic plate with voids in the cases when the heat conduction is modeled by the Fourier law and when it is modeled by the type III theory (with and without the inertial term). In all situations we show that, in general, the strong stability holds. In particular, we show slow decay of solutions for the Fourier case, that is, the solutions do not decay exponentially to zero (in general). However, if the coefficients satisfy a new relationship involving the inertial coefficient (singular case), we characterize the exponential decay of solutions. On the other hand, for the type III theory the situation is very different and we prove that generically the solutions decay to zero exponentially. This is another striking aspect when we compare both theories. This difference is a consequence of the couplings appearing in the type III case which are not present in the case of the Fourier law.

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Acknowledgements

The authors would like to thank the referees for their helpful suggestions and comments.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, CEP 36036-900, Brazil

    Hugo D. Fernández Sare

  2. Department of Mathematics, UPC, Colom 11, 08222, Terrassa, Spain

    Ramón Quintanilla

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  1. Hugo D. Fernández Sare
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Correspondence to Ramón Quintanilla.

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The work of R. Quintanilla is supported by the Ministerio de Ciencia, Innovación y Universidades under the research project “Análisis matemático aplicado a la termomecánica” (Ref. PID2019-105118GB-I00)

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Fernández Sare, H.D., Quintanilla, R. Porous-elastic Plates: Fourier Versus Type III. Appl Math Optim 84 (Suppl 1), 1055–1085 (2021). https://doi.org/10.1007/s00245-021-09793-5

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