Abstract
This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We here assume that the mass density is positive and the thermal conductivity tensor is positive definite. However, we do not assume the positivity of any other tensor. In this situation, we obtain Hölder continuous dependence results on the supply terms. We also sketch how to prove the continuous dependence on the initial data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ames K.A., Payne L.E.: Continuous dependence on initial-time geometry for a thermoelastic system with sign-indefinite elasticities. J. Math. Anal. Appl. 189, 693–714 (1995)
Ames K.A., Straughan B.: Continuous dependence results for initially prestressed thermoelastic bodies. Int. J. Eng. Sci. 30, 7–13 (1992)
Cimmelli V.A., dell’Isola F.: A moving boundary problem describing the growth of a droplet in its vapour. Arch. Mech. 45, 625–634 (1993)
dell’Isola F.: Linear growth of a liquid droplet divided from its vapour by a “soap bubble”-like fluid interface. Int. J. Eng. Sci. 27, 1053–1067 (1989)
Green A.E., Naghdi P.M.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15, 253–264 (1992)
Green A.E., Naghdi P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)
Green, A.E., Naghdi, P.M.: A unified procedure for construction of theories of deformable media. I. Classical continuum physics. Proc. R. Soc. Lond. A 448, 335–356, 357–377, 379–388 (1995)
Iesan D.: On the theory of thermoelasticity without energy dissipation. J. Therm. Stress. 21, 295–307 (1998)
Iesan D.: Thermopiezoelectricity without energy dissipation. Proc. R. Soc. Lond. A 464, 631–656 (2008)
Iesan D., Quintanilla R.: On the thermoelastic bodies with inner structure and microtemperatures. J. Math. Anal. Appl. 354, 12–23 (2009)
John F.: Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun. Pure Appl. Math. 13, 551–585 (1960)
Knops R.J., Payne L.E.: Improved estimates for continuous data dependence in linear elastodynamics. Math. Proc. Cambr. Philosoph. Soc. 103, 535–559 (1988)
Lazzari B., Nibbi R.: On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary. J. Math. Anal. Appl. 338, 317–329 (2008)
Leseduarte M.C., Magaña A., Quintanilla R.: On the time decay of solutions in porous-thermo-elasticity of type II. Discret. Continu. Dyn. Syst. Ser. B 13, 375–391 (2010)
Leseduarte M.C., Quintanilla R.: On uniqueness and continuous dependence in type III thermoelasticity. J. Math. Anal. Appl. 395, 429–436 (2012)
Liu Z., Quintanilla R.: Analiticity of solutions in the type III thermoelastic plates. IMA J. Appl. Math. 75, 356–365 (2010)
Liu Z., Quintanilla R.: Energy decay rates of mixed type II and type III thermoelastic system. Discret. Continu. Dyn. Syst. Ser. B 14, 1433–1444 (2010)
Liu Y., Lin C.: Phragmén–Lindelöf alternative and continuous dependence-type results for the thermoelasticity of type III. Appl. Anal.: Int. J. 83, 431–449 (2008)
Messaoudi S.A., Soufyane A.: Boundary stabilization of memory type in thermoelasticity of type III. Appl. Anal. 87, 13–28 (2008)
Puri P., Jordan P.M.: On the propagation of plane waves in type-III thermoelastic media. Proc. R. Soc. Lond. A 460, 3203–3221 (2004)
Qin Y., Deng S., Huang L., Ma Z., Su X.: Global existence for the three-dimensional thermoelastic equations of Type II. Q. Appl. Math. 68, 333–348 (2010)
Quintanilla R.: On the spatial behaviour in thermoelasticity without energy dissipation. J. Therm. Stress. 21, 213–224 (1999)
Quintanilla R.: Damping of end effects in a thermoelastic theory. Appl. Math. Lett. 14, 137–141 (2001)
Quintanilla R.: Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discret. Continu. Dyn. Syst. Ser. B 1, 463–470 (2001)
Quintanilla R.: Some remarks on growth and uniqueness in thermoelasticity. IJMMS 2003(10), 617–623 (2003)
Quintanilla R.: Convergence and structural stability in thermoelasticity. Appl. Math. Comp. 135, 287–300 (2003)
Quintanilla R.: Impossibility of localization in linear thermoelasticity. Proc. R. Soc. Lond. A 463, 3311–3322 (2007)
Quintanilla R., Racke R.: Stability in thermoelasticity of type III. Discret. Continu. Dyn. Syst. Ser. B 3, 383–400 (2003)
Quintanilla R., Straughan B.: Growth and uniqueness in thermoelasticity. Proc. R. Soc. Lond. A 456, 1419–1429 (2000)
Quintanilla R., Straughan B.: A note on discontinuity waves in type III thermoelasticity. Proc. R. Soc. Lond. A 460, 1169–1175 (2004)
Quintanilla R., Straughan B.: Energy bounds for some non-standard problems in thermoelasticity. Proc. R. Soc. Lond. A 461, 1147–1162 (2005)
Rionero S., Chirita S.: The Lagrange identity method in linear thermoelasticity. Int. J. Eng. Sci. 25, 935–947 (1987)
Wilkes N.S.: Continuous dependence and instability in linear thermoelasticity. SIAM J. Math. Anal. 11, 292–299 (1980)
Yang L., Wang Y.G.: Well-posedness and decay estimates for Cauchy problems of linear thermoelastic systems of type III in 3-D. Indiana Univ. Math. J. 55, 1333–1361 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
To G. A. Maugin in his 70th birthday.
Rights and permissions
About this article
Cite this article
Leseduarte, M.C., Quintanilla, R. Hölder stability in Type III thermoelastodynamics. Arch Appl Mech 84, 1465–1476 (2014). https://doi.org/10.1007/s00419-014-0827-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-014-0827-0