In this work we study modifications of the non-classical models of thermoelasticity, the one proposed Green and Lindsay and the one stated by Lord and Shulman, to two-temperature setting. We prove uniqueness results for the solutions of the systems of equations that model both theories for isotropic material. We also provide growth estimates for the solutions.
Dreyer, W. and Struchtrup, H.Heat pulse experiments revisited. Continuum Mechanics and Thermodynamics, 5, 3-50 (1993).
2.
Caviglia, G., Morro, A. and Straughan, B.Thermoelasticity at cryogenic temperatures . International Journal of Nonlinear Mechanics, 27, 251-263 (1992).
3.
Chandrasekharaiah, D.S.Thermoelasticity with second sound: A review . Applied Mechanics Review, 39, 355-376 ( 1986).
4.
Chandrasekharaiah, D.S.Hyperbolic thermoelasticity: A review of recent literature. Applied Mechanics Review, 51, 705-729 (1998).
5.
Hetnarski, R.B. and Ignaczak, J.Generalized thermoelasticity. Journal of Thermal Stresses, 22, 451-470 (1999).
6.
Green, A.E. and Naghdi, P.M.On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 15, 253-264 (1992).
7.
Green, A.E. and Naghdi, P.M.Thermoelasticity without energy dissipation. Journal of Elasticity , 31, 189-208 (1993).
8.
Batra, R.C.On heat conduction and wave propagation in non-simple rigid solids. Letters in Applied and Engineering Sciences, 3, 97-107 ( 1975).
9.
Youssef, H.M.Theory of two-temperature-generalized thermoelasticity . IMA Journal of Applied Mathematics, 71, 383-390 ( 2006).
10.
Green, A.E. and Lindsay, K.A.Thermoelasticity. Journal of Elasticity, 2, 1-7 ( 1972).
11.
Lord, H.W. and Shulman, Y.A generalized dynamical theory of thermoelasticity. Journal of Mechanics and Physics of Solids, 15, 299-309 (1967).
12.
Chen, P.J. and Gurtin, M.E.On a theory of heat involving two temperatures. Journal of Applied Mathematics and Physics (ZAMP), 19, 614-627 ( 1968).
13.
Chen, P.J. , Gurtin, M.E. and Williams, W.O.A note on non-simple heat conduction. Journal of Applied Mathematics and Physics (ZAMP), 19, 969-970 ( 1968).
14.
Chen, P.J. , Gurtin, M.E. and Williams, W.O.On the thermodynamics of non-simple materials with two temperatures. Journal of Applied Mathematics and Physics (ZAMP), 20, 107-112 (1969).
15.
Warren, W.E. and Chen, P.J.Wave propagation in two temperatures theory of thermoelaticity. Acta Mechanica, 16, 83-117 ( 1973).
16.
Puri, P. and Jordan, P.M.On the propagation of harmonic plane waves under the two-temperature theory . International Journal of Engineering Science, 44, 1113-1126 (2006).
17.
Quintanilla, R.Structural stability and continuous dependence of solutions in thermoelasticity of type III. Discrete and Continuous Dynamical Systems B, 1, 463-470 (2001).
18.
Quintanilla, R.On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mechanica, 168, 61-73 (2004).
19.
Racke, R.Thermoelasticity with second sound-exponential stability in linear and non-linear 1-D. Mathematical Methods in the Applied Sciences, 25, 409-441 (2002).
20.
Iesan, D. and Scalia, A.Thermoelastic Deformations, Kluwer, Dordrecht, 1996.
21.
Quintanilla, R. and Straughan, B.Growth and uniqueness in thermoelasticity . Proceedings of the Royal Society of London A, 456, 1419-1429 (2000).
22.
Quintanilla, R.Some remarks on growth and uniqueness in thermoelasticity . International Journal of Mathematics and Mathematical Sciences , 2003, 617-624 (2003).
23.
Ames, K.A. and Straughan, B.Non-standard and Improperly Posed Problems, Academic, San Diego, 1997.
24.
Gilbarg, D. and Trudinger, N.S.Elliptic Partial Differential Equations of Second Order, Springer, New York, 1977.
25.
Knops, R.J. and Payne, L.E.Growth estimates for solutions of evolutionary equations in Hilbert spaces with applications in elastodynamics. Archive for Rational Mechanics and Analysis, 41, 363-398 ( 1971).
