Thermoelastic Plates

  • Igor Chueshov
  • Irena Lasiecka
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter we study well-posedness and regularity issues associated with thermoelastic plates. The PDE model of thermoelasticity consists of coupled second-order (plate) and first-order (heat) equations. For this reason, treatment of thermoelastic models does not follow from the corresponding treatment of abstract second-order equations. A new functional framework needs to be developed.

In this chapter we undertake a study of solutions associated with von Karman evolution equations subject to thermal dissipation. We consider models with and without the rotational inertia terms.


Strong Solution Exponential Stability Continuous Semigroup Contraction Semigroup Resolvent Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 8.
    G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1–28.MathSciNetGoogle Scholar
  2. 9.
    G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic plates with free boundary conditions and without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155–182.MATHCrossRefMathSciNetGoogle Scholar
  3. 43.
    S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15–55.MATHMathSciNetGoogle Scholar
  4. 61.
    I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; see also
  5. 62.
    I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. Partial Diff. Eqs., 27 (2002), 1901–1951.MATHCrossRefMathSciNetGoogle Scholar
  6. 91.
    C. Dafermos, On the existence and asymptotic stability of solutions to to the equations of linear thermoelasticity, Arch. Ration. Mech. Anal., 29 (1968), 249–273.CrossRefMathSciNetGoogle Scholar
  7. 122.
    J. M. Ghidaglia, Some backward uniqueness results, Nonlin. Anal., 10 (1986), 777–790.MATHCrossRefMathSciNetGoogle Scholar
  8. 126.
    G. Giorgi, M.G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Diff. Eqs., 246 (2009) 3496–3517.MATHCrossRefMathSciNetGoogle Scholar
  9. 129.
    P. Grisvard, Caractérisation de quelques espaces d’interpolation, Arch. Ration. Mech. Anal., 25 (1967), 40–63.MATHCrossRefMathSciNetGoogle Scholar
  10. 139.
    D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.MATHGoogle Scholar
  11. 152.
    S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Chapman and Hall/CRC, Boca Raton, FL, 2000.MATHGoogle Scholar
  12. 159.
    T. Kato, Perturbation Theory of Linear Operators, Springer, New York, 1966.Google Scholar
  13. 165.
    H. Koch and I. Lasiecka, Backward uniqueness in linear thermo-elasticity with time and space variable coefficients. In: Functional Analysis and Evolution Equations. The Günter Lumer Volume, H. Amann et al., (Eds.), Birkhäuser, Basel, 2008, 389–405.CrossRefGoogle Scholar
  14. 169.
    I. Kukavica, Log-Log convexity and backward uniqueness, Proc. Amer. Math. Soc., 135 (2007), 2415–2421.MATHCrossRefMathSciNetGoogle Scholar
  15. 173.
    J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, 1989.MATHGoogle Scholar
  16. 178.
    J. Lagnese and J.L. Lions Modeling, Analysis and Control of Thin Plates, Masson, Paris, 1988.Google Scholar
  17. 207.
    I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the S.C. semigroup arising in abstract thermo-elastic equations, Adv. Diff. Eqs., 3 (1998), 387–416.MATHMathSciNetGoogle Scholar
  18. 208.
    I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free B.C., Annali Scuola Normale Superiore Pisa, 27 (1998), 457–482.MATHMathSciNetGoogle Scholar
  19. 216.
    I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Cambridge University Press, Cambridge, 2000.Google Scholar
  20. 218.
    I. Lasiecka, M. Renardy and R. Triggiani, Backward uniqueness of thermoelastic plates with rotational forces, Semigroup Forum, 62 (2001), 217–242.MATHMathSciNetGoogle Scholar
  21. 224.
    Z. Liu and M. Renardy, A note on the equation of thermoelastic plate, Appl. Math. Lett., 8 (1995), 1–6.CrossRefMathSciNetGoogle Scholar
  22. 241.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1986.Google Scholar
  23. 242.
    M. Potomkin, Asymptotic behaviour of solutions to nonlinear problem in thermoelasticity of plates, Rep. Nat. Acad. Sci. Ukraine, 2 (2009), 26–31, in Russian.Google Scholar

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© Springer Science+Business Media 2010

Authors and Affiliations

  1. 1.Department of Mechanics & MathematicsKharkov National UniversityKharkovUkraine
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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