A second gradient thermoelastic problem with the Coleman-Gurtin model

Noelia Bazarra; José R. Fernández; Víctor F. Prego; Ramón Quintanilla; Noelia Bazarra; José R. Fernández; Víctor F. Prego; Ramón Quintanilla

doi:10.3934/era.2026228

Electronic Research Archive

2026, Volume 34, Issue 8: 5144-5165. doi: 10.3934/era.2026228

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A second gradient thermoelastic problem with the Coleman-Gurtin model

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

Received: 08 April 2026 Revised: 02 June 2026 Accepted: 04 June 2026 Published: 16 June 2026

This article studies the Coleman-Gurtin thermoelastic model with second-gradient effects on the temperature from both analytical and numerical perspectives. Semigroup theory is used to obtain the existence, uniqueness, and continuous dependence of the solutions on the initial parameters. In the one-dimensional case, the polynomial decay of the solutions is obtained. The numerical analysis of the problem is then considered. Employing the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations of a weak form of the thermoelastic problem are introduced. A discrete stability property and an a priori error analysis are also provided. Finally, some numerical simulations are presented to show the numerical behavior of the approximations and the discrete energy.
- Coleman-Gurtin model,
- second gradient,
- existence and uniqueness,
- stability and energy decay,
- finite elements,
- error estimates
Citation: Noelia Bazarra, José R. Fernández, Víctor F. Prego, Ramón Quintanilla. A second gradient thermoelastic problem with the Coleman-Gurtin model[J]. Electronic Research Archive, 2026, 34(8): 5144-5165. doi: 10.3934/era.2026228

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Abstract

This article studies the Coleman-Gurtin thermoelastic model with second-gradient effects on the temperature from both analytical and numerical perspectives. Semigroup theory is used to obtain the existence, uniqueness, and continuous dependence of the solutions on the initial parameters. In the one-dimensional case, the polynomial decay of the solutions is obtained. The numerical analysis of the problem is then considered. Employing the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations of a weak form of the thermoelastic problem are introduced. A discrete stability property and an a priori error analysis are also provided. Finally, some numerical simulations are presented to show the numerical behavior of the approximations and the discrete energy.

References

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