Skip to main content
SpringerLink
Log in

On the final boundary value problems in linear thermoelasticity

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

In the present study we derive some uniqueness criteria for solutions of the Cauchy problem for the standard equations of dynamical linear thermoelasticity backward in time. We use Lagrange-Brun identities combined with some differential inequalities in order to show that the final boundary value problem associated with the linear thermoelasticity backward in time has at most one solution in appropriate classes of displacement-temperature fields. The uniqueness results are obtained under the assumptions that the density mass and the specific heat are strictly positive and the conductivity tensor is positive definite.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brun L (1965) Sur l’unicité en thermoelasticité dynamique et diverses expressions analogues à la formule de Clapeyron. C R Acad Sci Paris 261:2584–2587

    MathSciNet  Google Scholar 

  2. Brun L (1969) Méthodes énergétiques dans les systèmes évolutifs linéaires, Première partie : Separation des énergies, Deuxième partie : Théorèmes d’unicité. J Méc 8:125–166. 167–192

    MathSciNet  MATH  Google Scholar 

  3. Knops RJ, Payne LE (1970) On uniqueness and continuous data dependence in dynamical problems of linear thermoelasticity. Int J Solids Struct 6:1173–1184

    Article  MATH  Google Scholar 

  4. Levine HA (1970) On a theorem of Knops and Payne in dynamical thermoelasticity. Arch Ration Mech Anal 38:290–307

    Article  MATH  Google Scholar 

  5. Wilkes NS (1980) Continuous dependence and instability in linear thermoelasticity. SIAM J Appl Math 11:292–299

    Article  MathSciNet  MATH  Google Scholar 

  6. Rionero S, Chiriţă S (1987) The Lagrange identity method in linear thermoelasticity. Int J Eng Sci 25:935–947

    Article  MATH  Google Scholar 

  7. Ieşan D (1989) On some theorems in thermoelastodynamics. Rev Roum Sci Tech, Sér Méc Appl 34:101–111

    MATH  Google Scholar 

  8. Ieşan D (1989) Reciprocity, uniqueness and minimum principles in the dynamic theory of thermoelasticity. J Therm Stresses 12:465–482

    Article  Google Scholar 

  9. Ames KA, Payne LE (1991) Stabilizing solutions of the equations of dynamical linear thermoelasticity backward in time. Stab Appl Anal Contin Media 1:243–260

    MathSciNet  Google Scholar 

  10. Ciarletta M, Chiriţă S (2001) Spatial behavior in linear thermoelasticity backward in time. In: Chao CK, Lin CY (eds) Proceedings of the fourth international congress on thermal stresses, Osaka, Japan, pp 485–488

  11. Ciarletta M, Chiriţă S (2002) Asymptotic partition in the linear thermoelasticity backward in time. In: Mathematical models and methods for smart materials, Cortona, 2001. Series on advances in mathematics for applied sciences, vol 62. World Scientific, River Edge, pp 31–41

    Chapter  Google Scholar 

  12. Ciarletta M (2002) On the uniqueness and continuous dependence of solutions in dynamical thermoelasticity backward in time. J Therm Stresses 25:969–984

    Article  MathSciNet  Google Scholar 

  13. Iovane G, Passarella F (2004) Spatial behaviour in dynamical thermoelasticity backward in time for porous media. J Therm Stresses 27:97–109

    Article  MathSciNet  Google Scholar 

  14. Passarella F, Tibullo V (2010) Some results in linear theory of thermoelasticity backward in time for microstretch materials. J Therm Stresses 33:559–576

    Article  Google Scholar 

  15. Koch H, Lasiecka I (2007) Backward uniqueness in linear thermoelasticity with time and space variable coefficients. In: Functional analysis and evolution equations: Gunter Lumer volume. Birkhäuser, Basel, pp 389–403

    Google Scholar 

  16. Ames KA, Payne LE (1998) Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation. Math Models Methods Appl Sci 8:187–202

    Article  MathSciNet  MATH  Google Scholar 

  17. Ames KA, Payne LE, Schaefer PW (2004) On a nonstandard problem for heat conduction in a cylinder. Appl Anal 83:125–133

    Article  MathSciNet  MATH  Google Scholar 

  18. Ames KA, Payne LE, Schaefer PW (2004) Energy and pointwise bounds in some nonstandard parabolic problems. Proc R Soc Edinb A 134:1–9

    Article  MathSciNet  MATH  Google Scholar 

  19. Ames KA, Payne LE, Song JC (2005) On two classes of nonstandard parabolic problems. J Math Anal Appl 311:254–267

    Article  MathSciNet  MATH  Google Scholar 

  20. Song JC (2001) Spatial decay for solutions of Cauchy problems for perturbed heat equations. Math Models Methods Appl Sci 11:797–808

    Article  MathSciNet  MATH  Google Scholar 

  21. Payne LE, Schaefer PW, Song JC (2004) Improved bounds for some nonstandard problems in generalized heat conduction. J Math Anal Appl 298:325–340

    Article  MathSciNet  MATH  Google Scholar 

  22. Payne LE, Schaefer PW, Song JC (2004) Some nonstandard problems in viscous flow. Math Methods Appl Sci 27:2045–2053

    Article  MathSciNet  MATH  Google Scholar 

  23. Quintanilla R, Straughan B (2005) Bounds for some nonstandard problems in porous viscous flow and viscous Green–Naghdi fluids. Proc R Soc Lond A 461:3159–3168

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Chiriţă S (2009) On some non-standard problems in linear thermoelasticity. J Therm Stresses 32:1256–1269

    Article  Google Scholar 

  25. Chiriţă S, Ciarletta M (2010) Spatial behavior for some non-standard problems in linear thermoelasticity without energy dissipation. J Math Anal Appl 367:58–68

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, “Al. I. Cuza” University of Iaşi, Blvd. Carol I, no. 11, 700506, Iaşi, Romania

    Stan Chiriţă

  2. “Octav Mayer” Mathematics Institute, Romanian Academy of Science, Iaşi Branch, Bd. Carol I, no. 8, 700506, Iaşi, Romania

    Stan Chiriţă

Authors
  1. Stan Chiriţă
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stan Chiriţă.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chiriţă, S. On the final boundary value problems in linear thermoelasticity. Meccanica 47, 2005–2011 (2012). https://doi.org/10.1007/s11012-012-9570-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-012-9570-1

Keywords

Search

Navigation