Study of the energy decay for three second-gradient thermoelastic models

Noelia Bazarra; José R. Fernández; Marta Pellicer; Ramón Quintanilla; Noelia Bazarra; José R. Fernández; Marta Pellicer; Ramón Quintanilla

doi:10.3934/era.2026284

Electronic Research Archive

2026, Volume 34, Issue 9: 6483-6512. doi: 10.3934/era.2026284

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Research article

Study of the energy decay for three second-gradient thermoelastic models

1.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, Vigo 36310, Spain
2.
Departament d'Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, C. Maria Aurèlia Capmany, Girona, Catalunya, Spain
3.
Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain

Received: 22 May 2026 Revised: 09 July 2026 Accepted: 13 July 2026 Published: 24 July 2026

We study a class of one-dimensional thermoelastic models combining second-gradient effects, both in the mechanical displacement and the thermal component, as dissipation mechanisms. While second-gradient elasticity is always assumed in the mechanical part, three alternative thermal models are considered, involving different combinations of thermal displacement and temperature gradients. For each system, we establish existence and uniqueness of solutions using semigroup theory and analyze their long-time behavior. We show that exponential stability generally fails in most of the cases and that all solutions exhibit polynomial decay toward equilibrium, with rate of order $ t^{-1/2} $. The decay analysis relies on resolvent-based semigroup techniques and, in a degenerate case, on energy methods. In addition, numerical approximations are investigated for two of the models. A priori error estimates for a fully discrete scheme are derived, and numerical experiments are presented to illustrate the accuracy of the method and to confirm the theoretical decay rates.
- second gradient,
- existence and uniqueness,
- polynomial energy decay,
- finite elements,
- error estimates
Citation: Noelia Bazarra, José R. Fernández, Marta Pellicer, Ramón Quintanilla. Study of the energy decay for three second-gradient thermoelastic models[J]. Electronic Research Archive, 2026, 34(9): 6483-6512. doi: 10.3934/era.2026284

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Abstract

We study a class of one-dimensional thermoelastic models combining second-gradient effects, both in the mechanical displacement and the thermal component, as dissipation mechanisms. While second-gradient elasticity is always assumed in the mechanical part, three alternative thermal models are considered, involving different combinations of thermal displacement and temperature gradients. For each system, we establish existence and uniqueness of solutions using semigroup theory and analyze their long-time behavior. We show that exponential stability generally fails in most of the cases and that all solutions exhibit polynomial decay toward equilibrium, with rate of order $ t^{-1/2} $. The decay analysis relies on resolvent-based semigroup techniques and, in a degenerate case, on energy methods. In addition, numerical approximations are investigated for two of the models. A priori error estimates for a fully discrete scheme are derived, and numerical experiments are presented to illustrate the accuracy of the method and to confirm the theoretical decay rates.

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