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On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation

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Abstract

It is known that in the case that several constitutive tensors fail to be positive definite the system of the thermoelasticity could become unstable and, in certain cases, ill-posed in the sense of Hadamard. In this paper, we consider the Moore–Gibson–Thompson thermoelasticity in the case that some of the constitutive tensors fail to be positive and we will prove basic results concerning uniqueness and instability of solutions. We first consider the case of the heat conduction when dissipation condition holds, but some constitutive tensors can fail to be positive. In this case, we prove the uniqueness and instability by means of the logarithmic convexity argument. Second we study the thermoelastic system only assuming that the thermal conductivity tensor and the mass density are positive and we obtain the uniqueness of solutions by means of the Lagrange identities method. By the logarithmic convexity argument we prove later the instability of solutions whenever the elasticity tensor fails to be positive, but assuming that the conductivity rate is positive and the thermal dissipation condition hold. We also sketch similar results when conductivity rate and/or the thermal conductivity fail to be positive definite, but the elasticity tensor is positive definite and the dissipation condition holds. Last sections are devoted to considering the case when a third-order equation is proposed for the displacement (which comes from the viscoelasticiy). A similar study is sketched in these cases.

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Notes

  1. This is motivated becase type III heat conduction violates the principle of causality see [10, 32].

  2. However it is worth recalling that the basic axioms of the thermomechanics imply that the tensor \(k_{ij}\) is semi-definite positive.

  3. This kind of relaxation function satisfies the usual requirements of fading memory for viscoelastic materials.

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Acknowledgements

Research supported by project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness. This work is also part of the project “Análisis Matemático Aplicado a la Termomecánica” which is submitted to the Spanish Ministry of Science, Innovation and Universities. M. Pellicer is part of the Catalan Research group 2017 SGR 1392 and has been supported by the MINECO Grant MTM2017-84214-C2-2-P (Spain). The authors would also like to thank the anonymous reviewer for her or his useful comments, that has allowed us to improve this contribution.

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  1. Departament d’Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, C. Maria Aurèlia Capmany, 41 (Campus Montilivi), 17003, Girona, Catalunya, Spain

    Marta Pellicer

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, C. Colom 11, 08222, Terrassa, Barcelona, Catalunya, Spain

    Ramon Quintanilla

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  1. Marta Pellicer
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Pellicer, M., Quintanilla, R. On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation. Z. Angew. Math. Phys. 71, 84 (2020). https://doi.org/10.1007/s00033-020-01307-7

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