# zbMATH — the first resource for mathematics

Energy bounds for some non-standard problems in thermoelasticity. (English) Zbl 1145.74337
Summary: A. E. Green and P. M. Naghdi developed two theories of thermoelasticity, called type II and type III, which are likely to be more natural candidates for the identification of a thermoelastic body than the usual theory. We here derive energy bounds for a class of problem in which the ’initial data’ are given as a combination of data at time $$t=0$$ and at a later time $$t=T$$. Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time.

##### MSC:
 74F05 Thermal effects in solid mechanics 35B45 A priori estimates in context of PDEs 35Q72 Other PDE from mechanics (MSC2000) 74B20 Nonlinear elasticity 74H55 Stability of dynamical problems in solid mechanics
Full Text:
##### References:
 [1] Ames, K.A. & Payne, L.E. 1991 Stabilizing solutions of the equations of dynamical linear thermoelasticity backward in time. Stab. Appl. Anal. Cont. Media 1, 243–260. [2] Ames, K.A. & Payne, L.E. 1994 Uniqueness and continuous dependence of solutions to a multidimensional thermoelastic contact problem. J. Elasticity 34, 139–148. · Zbl 0809.73055 [3] Ames, K.A. & Payne, L.E. 1995 Continuous dependence on initial-time geometry for a thermoelastic system with sign-indefinite elasticities. J. Math. Anal. Appl. 189, 693–714. · Zbl 0857.35128 [4] Ames, K.A. & Payne, L.E. 1999 Continuous dependence on modelling for some well-posed perturbations of the backward heat equation. J. Inequal. Appl. 3, 51–64. · Zbl 0928.35015 [5] Ames, K.A. & Straughan, B. 1997 Non-standard and improperly posed problems. Mathematics in science and engineering series, vol. 194. California: Academic Press. [6] Ames, K.A., Clark, G.W., Epperson, J.F. & Oppenheimer, S.F. 1998 A comparison of regularizations for an ill-posed problem. Math. Comput. 67, 1451–1471. · Zbl 0898.35110 [7] Ames, K.A., Payne, L.E. & Schaefer, P.W. 2004 Energy and pointwise bounds in some nonstandard parabolic problems. Proc. R. Soc. Edinb. A 134, 1–9. · Zbl 1056.35104 [8] Ames, K.A., Payne, L.E. & Schaefer, P.W. 2004 On a nonstandard problem for heat conduction in a cylinder. Appl. Anal. 83, 125–133. · Zbl 1052.35066 [9] Chandrasekharaiah, D.S. 1996 A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation. J. Elasticity 43, 279–283. · Zbl 0876.73014 [10] Chandrasekharaiah, D.S. 1998 Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729. [11] Green, A.E. & Naghdi, P.M. 1991 A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A 432, 171–194. · Zbl 0726.73004 [12] Green, A.E. & Naghdi, P.M. 1992 On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253–264. [13] Green, A.E. & Naghdi, P.M. 1993 Thermoelasticity without energy-dissipation. J. Elasticity 31, 189–208. [14] Green, A.E. & Naghdi, P.M. 1995 A unified procedure for construction of theories of deformable media. I. Classical continuum mechanics. Proc. R. Soc. A 448, 335–356. · Zbl 0868.73013 [15] Green, A.E. & Naghdi, P.M. 1995 A unified procedure for construction of theories of deformable media. II. Generalised continua. Proc. R. Soc. A 448, 357–377. · Zbl 0868.73013 [16] Green, A.E. & Naghdi, P.M. 1995 A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua. Proc. R. Soc. A 448, 379–388. · Zbl 0868.73013 [17] Hetnarski, R.B. & Ignaczak, I. 1999 Generalised thermoelasticity. J. Therm. Stresses 22, 451–476. [18] Iesan, D. 1998 On the theory of thermoelasticity without energy dissipation. J. Therm. Stresses 21, 295–307. [19] Nappa, L. 1998 Spatial decay estimates for the evolution equations of linear thermoelasticity without energy dissipation. J. Therm. Stresses 21, 581–592. [20] Quintanilla, R. & Straughan, B. 2000 Growth and uniqueness in thermoelasticity. Proc. R. Soc. A 456, 1419–1429. · Zbl 0985.74018 [21] Payne, L.E. & Schaefer, P.W. 2002 Energy bounds for some nonstandard problems in partial differential equations. J. Math. Anal. Appl. 273, 75–92. · Zbl 1121.35331 [22] Payne, L.E., Schaefer, P.W. & Song, J.C. In press Some nonstandard problems in generalized heat conduction. ZAMP · Zbl 1080.35034 [23] Quintanilla, R. 2001 Damping of end effects in a thermoelastic theory. Appl. Math. Lett. 14, 137–141. · Zbl 0971.74037 [24] Quintanilla, R. 2001 Structural stability and continuous dependence of solutions in thermoelasticity of type III. Discr. Cont. Dyn. Syst. B 1, 463–470. · Zbl 1010.74016 [25] Quintanilla, R. 2002 On existence in thermoelasticity without energy dissipation. J. Therm. Stresses 25, 195–202. [26] Quintanilla, R. 2003 Convergence and structural stability in thermoelasticity. Appl. Math. Comput. 135, 287–300. [27] Quintanilla, R. & Racke, R. 2003 Stability in thermoelasticity of type III. Discr. Cont. Dyn. Syst. B 3, 383–400. · Zbl 1118.74012 [28] Quintanilla, R. & Straughan, B. 2002 Explosive instabilities in heat transmission. Proc. R. Soc. A 458, 2833–2837. · Zbl 1047.80002 [29] Quintanilla, R. & Straughan, B. 2004 Discontinuity waves in type III thermoelasticity. Proc. R. Soc. A 460, 1169–1175. · Zbl 1070.74024 [30] Rionero, S. & Chirita, S. 1987 The Lagrange identity method in linear thermoelasticity. Int. J. Eng. Sci. 25, 935–947. · Zbl 0617.73007 [31] Zhang, X. & Zuazua, E. 2003 Decay of solutions of the system of thermoelasticity of type III. Commun. Contemp. Math. 5, 25–84. · Zbl 1136.74318
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.