Uniqueness and exponential instability in a new two-temperature thermoelastic theory

José R. Fernández; Ramón Quintanilla; José R. Fernández; Ramón Quintanilla

doi:10.3934/math.2021321

AIMS Mathematics

2021, Volume 6, Issue 6: 5440-5451. doi: 10.3934/math.2021321

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Research article

Uniqueness and exponential instability in a new two-temperature thermoelastic theory

José R. Fernández ¹,
Ramón Quintanilla ^{2
,
,}

1.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
2.
Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain

Received: 28 January 2021 Accepted: 11 March 2021 Published: 15 March 2021
MSC : 35Q74, 74H10, 74H25, 80A19, 74F05

In this work we consider the temperature-rate dependent two temperatures thermoelastic theory. It has been proposed very recently. We study the case in which the elasticity tensor may not be positive definite. Thus, the problem can be ill posed in the sense of Hadamard. We adapt the logarithmic convexity argument to the specific situation proposed by this theory. That is, we define a suitable function on the solutions satisfying that the logarithm is convex. Uniqueness and instability of the solutions under suitable conditions on the constitutive tensors are proved.
- temperature-rate dependent two-temperature thermoelasticity,
- uniqueness,
- logarithmic convexity,
- exponential instability
Citation: José R. Fernández, Ramón Quintanilla. Uniqueness and exponential instability in a new two-temperature thermoelastic theory[J]. AIMS Mathematics, 2021, 6(6): 5440-5451. doi: 10.3934/math.2021321

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Abstract

In this work we consider the temperature-rate dependent two temperatures thermoelastic theory. It has been proposed very recently. We study the case in which the elasticity tensor may not be positive definite. Thus, the problem can be ill posed in the sense of Hadamard. We adapt the logarithmic convexity argument to the specific situation proposed by this theory. That is, we define a suitable function on the solutions satisfying that the logarithm is convex. Uniqueness and instability of the solutions under suitable conditions on the constitutive tensors are proved.

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