May  2014, 19(3): 679-695. doi: 10.3934/dcdsb.2014.19.679

On the backward in time problem for the thermoelasticity with two temperatures

1. 

Matemàtica Aplicada 2, ETSEIAT, Universitat Politécnica de Catalunya, Colom, 11. Terrassa, 08222, Barcelona, Spain, Spain

Received  December 2012 Revised  July 2013 Published  February 2014

This paper is devoted to the study of the existence, uniqueness, continuous dependence and spatial behaviour of the solutions for the backward in time problem determined by the Type III with two temperatures thermoelastodynamic theory. We first show the existence, uniqueness and continuous dependence of the solutions. Instability of the solutions for the Type II with two temperatures theory is proved later. For the one-dimensional Type III with two temperatures theory, the exponential instability is also pointed-out. We also analyze the spatial behaviour of the solutions. By means of the exponentially weighted Poincaré inequality, we are able to obtain a function that defines a measure on the solutions and, therefore, we obtain the usual exponential type alternative for the solutions of the problem defined in a semi-infinite cylinder.
Citation: M. Carme Leseduarte, Ramon Quintanilla. On the backward in time problem for the thermoelasticity with two temperatures. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 679-695. doi: 10.3934/dcdsb.2014.19.679
References:
[1]

E. S. Awad, A note on the spatial decay estimates in non-classical linear thermoelastic semi-cylindrical bounded domains,, J. Thermal Stresses, 34 (2011), 147. doi: 10.1080/01495739.2010.511942. Google Scholar

[2]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures,, Jour. Appl. Math. Phys. (ZAMP), 19 (1968), 614. doi: 10.1007/BF01594969. Google Scholar

[3]

P. J. Chen, M. E. Gurtin and W. O. Williams, A note on non-simple heat conduction,, Jour. Appl. Math. Phys. (ZAMP), 19 (1968), 969. doi: 10.1007/BF01602278. Google Scholar

[4]

P. J. Chen, M. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures,, Jour. Appl. Math. Phys. (ZAMP), 20 (1969), 107. doi: 10.1007/BF01591120. Google Scholar

[5]

J. I. Díaz and R. Quintanilla, Spatial and contiuous dependence estimates in linear viscoelastity,, J. Math. Anal. Appl., 273 (2002), 1. doi: 10.1016/S0022-247X(02)00200-7. Google Scholar

[6]

M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Lett., 22 (2009), 1374. doi: 10.1016/j.aml.2009.03.010. Google Scholar

[7]

A. S. El-Karamany and M. A. Ezzat, On the two-temperature Green-Naghdi thermoelasticity theories,, J. Thermal Stresses, 34 (2011), 1207. doi: 10.1080/01495739.2011.608313. Google Scholar

[8]

J. N. Flavin, R. J. Knops and L. E. Payne, Decay estimates for the constrained elastic cylinder of variable cross-section,, Quart. Appl. Math., 47 (1989), 325. Google Scholar

[9]

J. N. Flavin, R. J. Knops and L. E. Payne, Energy bounds in dynamical problems for a semi-infinite elastic beam,, in Elasticity: Mathematical Methods and Applications (eds. G. Eason and R.W. Ogden), (1989), 101. Google Scholar

[10]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford: Oxford Mathematical Monographs, (1985). Google Scholar

[11]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253. doi: 10.1080/01495739208946136. Google Scholar

[12]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189. doi: 10.1007/BF00044969. Google Scholar

[13]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media, I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua, Proc., Roy. Soc. London A, 448 (1995), 379. Google Scholar

[14]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity,, J. Thermal Stresses, 22 (1999), 451. doi: 10.1080/014957399280832. Google Scholar

[15]

R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity,, Int. J. Solids Struct., 37 (1999), 215. doi: 10.1016/S0020-7683(99)00089-X. Google Scholar

[16]

C. O. Horgan, L. E. Payne and L. T. Wheeler, Spatial decay estimates in transient heat conduction,, Quart. Appl. Math., 42 (1984), 119. Google Scholar

[17]

C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects in functionally graded heat conducting materials,, Quart. Appl. Math., 59 (2001), 529. Google Scholar

[18]

D. Iesan, On the theory of thermoelasticity without energy dissipation,, J. Thermal Stresses, 21 (1998), 295. doi: 10.1080/01495739808956148. Google Scholar

[19]

D. Iesan, Thermopiezoelectricity without energy dissipation,, Proc. Roy. Soc. London A, 464 (2008), 631. doi: 10.1098/rspa.2007.0264. Google Scholar

[20]

D. Iesan and R. Quintanilla, On the thermoelastic bodies with inner structure and microtemperatures,, J. Math. Anal. Appl., 354 (2009), 12. doi: 10.1016/j.jmaa.2008.12.017. Google Scholar

[21]

J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds,, Oxford: Oxford Mathematical Monographs, (2010). Google Scholar

[22]

B. Lazzari and R. Nibbi, On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary,, J. Math. Anal. Appl., 338 (2008), 317. doi: 10.1016/j.jmaa.2007.05.017. Google Scholar

[23]

M. C. Leseduarte, A. Magaña and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II,, Discrete Contin. Dyn. Sys., 13 (2010), 375. doi: 10.3934/dcdsb.2010.13.375. Google Scholar

[24]

M. C. Leseduarte and R. Quintanilla, On uniqueness and continuous dependence in type III thermoelasticity,, J. Math. Anal. Appl., 395 (2012), 429. doi: 10.1016/j.jmaa.2012.05.019. Google Scholar

[25]

M. C. Leseduarte and R. Quintanilla, Phragmén-Lindelöf alternative for an exact heat conduction equation with delay,, Commun. Pure Appl. Anal., 12 (2013), 1221. doi: 10.3934/cpaa.2013.12.1221. Google Scholar

[26]

Y. Liu and C. Lin, Phragmén-Lindelöf alternative and continuous dependence-type results for the thermoelasticity of type III,, Appl. Anal., 87 (2008), 431. doi: 10.1080/00036810801927963. Google Scholar

[27]

Z. Liu and R. Quintanilla, Energy decay rates of mixed type II and type III thermoelastic system,, Discrete Contin. Dyn. Syst., 14 (2010), 1433. doi: 10.3934/dcdsb.2010.14.1433. Google Scholar

[28]

Z. Liu and S. Zheng, Semigroups Associated to Dissipative Systems,, Chapman & Hall/CRC Boca Raton, (1999). Google Scholar

[29]

J. C. Maxwell, Theory of Heat,, Cambridge University Press, (2011). doi: 10.1017/CBO9781139057943. Google Scholar

[30]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of memory type in thermoelasticity of type III,, Appl. Anal., 87 (2008), 13. doi: 10.1080/00036810701714180. Google Scholar

[31]

S. Mukhopadyay, R. Prasad and R. Kumar, On the theory of two-temeperature thermoelasticity with two phase-lags,, J. Thermal Stresses, 34 (2011), 352. doi: 10.1080/01495739.2010.550815. Google Scholar

[32]

P. Puri and P. M. Jordan, On the propagation of plane waves in type-III thermoelastic media,, Proc. Roy. Soc. London A, 460 (2004), 3203. doi: 10.1098/rspa.2004.1341. Google Scholar

[33]

Y. Qin, S. Deng, L. Huang, Z. Ma and X. Su, Global existence for the three-dimensional thermoelastic equations of Type II,, Quart. Appl. Math., 68 (2010), 333. Google Scholar

[34]

R. Quintanilla, On the spatial behaviour in thermoelasticity without energy dissipation,, J. Thermal Stresses, 22 (1999), 213. doi: 10.1080/014957399280977. Google Scholar

[35]

R. Quintanilla, Damping of end effects in a thermoelastic theory,, Appl. Math. Lett., 14 (2001), 137. doi: 10.1016/S0893-9659(00)00125-7. Google Scholar

[36]

R. Quintanilla, Convergence and structural stability in thermoelasticity,, Appl. Math. Comput., 135 (2003), 287. doi: 10.1016/S0096-3003(01)00331-9. Google Scholar

[37]

R. Quintanilla, Exponential stability and uniqueness in thermoelasticity with two temperatures,, Dyn. Conti. Discrete Impuls. Syst. Ser. A: Math. Anal., 11 (2004), 57. Google Scholar

[38]

R. Quintanilla, On the impossibility of localization in linear thermoelasticity,, Proc. Roy. Soc. London A, 463 (2007), 3311. doi: 10.1098/rspa.2007.0076. Google Scholar

[39]

R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction,, J. Thermal Stresses, 31 (2008), 260. doi: 10.1080/01495730701738272. Google Scholar

[40]

R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction,, J. Thermal Stresses, 32 (2009), 1270. doi: 10.1080/01495730903310599. Google Scholar

[41]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type III,, Discrete Contin. Dyn. Syst., 3 (2003), 383. doi: 10.3934/dcdsb.2003.3.383. Google Scholar

[42]

R. Quintanilla and G. Saccomandi, Phragmén-Lindelöf alternative of exponential type for the solutions of a fourth order dispersive equation,, Rend. Lincei Mat. Appl., 23 (2012), 105. doi: 10.4171/RLM/620. Google Scholar

[43]

R. Quintanilla and B. Straughan, Growth and uniqueness in thermoelasticity,, Proc. Roy. Soc. London A, 456 (2000), 1419. doi: 10.1098/rspa.2000.0569. Google Scholar

[44]

R. Quintanilla and B. Straughan, Energy bounds for some non-standard problems in thermoelasticity,, Proc. Roy. Soc. London A, 461 (2005), 1147. doi: 10.1098/rspa.2004.1381. Google Scholar

[45]

S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model,, J. Thermal Stresses, 30 (2007), 231. doi: 10.1080/01495730601130919. Google Scholar

[46]

B. Straughan, Heat Waves,, Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4614-0493-4. Google Scholar

[47]

D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales,, ASME J. Heat Transfer, 117 (1995), 8. doi: 10.1115/1.2822329. Google Scholar

[48]

W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelasticity,, Acta Mechanica, 16 (1973), 21. doi: 10.1007/BF01177123. Google Scholar

[49]

L. Yang and Y. G. Wang, Well-posedness and decay estimates for Cauchy problems of linear thermoelastic systems of type III in 3-D,, Indiana Univ. Math. J., 55 (2006), 1333. doi: 10.1512/iumj.2006.55.2799. Google Scholar

[50]

H. M. Youssef, Theory of two-temperature thermoelasticity without energy dissipation,, J. Thermal Stresses, 34 (2011), 138. doi: 10.1080/01495739.2010.511941. Google Scholar

show all references

References:
[1]

E. S. Awad, A note on the spatial decay estimates in non-classical linear thermoelastic semi-cylindrical bounded domains,, J. Thermal Stresses, 34 (2011), 147. doi: 10.1080/01495739.2010.511942. Google Scholar

[2]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures,, Jour. Appl. Math. Phys. (ZAMP), 19 (1968), 614. doi: 10.1007/BF01594969. Google Scholar

[3]

P. J. Chen, M. E. Gurtin and W. O. Williams, A note on non-simple heat conduction,, Jour. Appl. Math. Phys. (ZAMP), 19 (1968), 969. doi: 10.1007/BF01602278. Google Scholar

[4]

P. J. Chen, M. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures,, Jour. Appl. Math. Phys. (ZAMP), 20 (1969), 107. doi: 10.1007/BF01591120. Google Scholar

[5]

J. I. Díaz and R. Quintanilla, Spatial and contiuous dependence estimates in linear viscoelastity,, J. Math. Anal. Appl., 273 (2002), 1. doi: 10.1016/S0022-247X(02)00200-7. Google Scholar

[6]

M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Lett., 22 (2009), 1374. doi: 10.1016/j.aml.2009.03.010. Google Scholar

[7]

A. S. El-Karamany and M. A. Ezzat, On the two-temperature Green-Naghdi thermoelasticity theories,, J. Thermal Stresses, 34 (2011), 1207. doi: 10.1080/01495739.2011.608313. Google Scholar

[8]

J. N. Flavin, R. J. Knops and L. E. Payne, Decay estimates for the constrained elastic cylinder of variable cross-section,, Quart. Appl. Math., 47 (1989), 325. Google Scholar

[9]

J. N. Flavin, R. J. Knops and L. E. Payne, Energy bounds in dynamical problems for a semi-infinite elastic beam,, in Elasticity: Mathematical Methods and Applications (eds. G. Eason and R.W. Ogden), (1989), 101. Google Scholar

[10]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford: Oxford Mathematical Monographs, (1985). Google Scholar

[11]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253. doi: 10.1080/01495739208946136. Google Scholar

[12]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189. doi: 10.1007/BF00044969. Google Scholar

[13]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media, I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua, Proc., Roy. Soc. London A, 448 (1995), 379. Google Scholar

[14]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity,, J. Thermal Stresses, 22 (1999), 451. doi: 10.1080/014957399280832. Google Scholar

[15]

R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity,, Int. J. Solids Struct., 37 (1999), 215. doi: 10.1016/S0020-7683(99)00089-X. Google Scholar

[16]

C. O. Horgan, L. E. Payne and L. T. Wheeler, Spatial decay estimates in transient heat conduction,, Quart. Appl. Math., 42 (1984), 119. Google Scholar

[17]

C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects in functionally graded heat conducting materials,, Quart. Appl. Math., 59 (2001), 529. Google Scholar

[18]

D. Iesan, On the theory of thermoelasticity without energy dissipation,, J. Thermal Stresses, 21 (1998), 295. doi: 10.1080/01495739808956148. Google Scholar

[19]

D. Iesan, Thermopiezoelectricity without energy dissipation,, Proc. Roy. Soc. London A, 464 (2008), 631. doi: 10.1098/rspa.2007.0264. Google Scholar

[20]

D. Iesan and R. Quintanilla, On the thermoelastic bodies with inner structure and microtemperatures,, J. Math. Anal. Appl., 354 (2009), 12. doi: 10.1016/j.jmaa.2008.12.017. Google Scholar

[21]

J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds,, Oxford: Oxford Mathematical Monographs, (2010). Google Scholar

[22]

B. Lazzari and R. Nibbi, On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary,, J. Math. Anal. Appl., 338 (2008), 317. doi: 10.1016/j.jmaa.2007.05.017. Google Scholar

[23]

M. C. Leseduarte, A. Magaña and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II,, Discrete Contin. Dyn. Sys., 13 (2010), 375. doi: 10.3934/dcdsb.2010.13.375. Google Scholar

[24]

M. C. Leseduarte and R. Quintanilla, On uniqueness and continuous dependence in type III thermoelasticity,, J. Math. Anal. Appl., 395 (2012), 429. doi: 10.1016/j.jmaa.2012.05.019. Google Scholar

[25]

M. C. Leseduarte and R. Quintanilla, Phragmén-Lindelöf alternative for an exact heat conduction equation with delay,, Commun. Pure Appl. Anal., 12 (2013), 1221. doi: 10.3934/cpaa.2013.12.1221. Google Scholar

[26]

Y. Liu and C. Lin, Phragmén-Lindelöf alternative and continuous dependence-type results for the thermoelasticity of type III,, Appl. Anal., 87 (2008), 431. doi: 10.1080/00036810801927963. Google Scholar

[27]

Z. Liu and R. Quintanilla, Energy decay rates of mixed type II and type III thermoelastic system,, Discrete Contin. Dyn. Syst., 14 (2010), 1433. doi: 10.3934/dcdsb.2010.14.1433. Google Scholar

[28]

Z. Liu and S. Zheng, Semigroups Associated to Dissipative Systems,, Chapman & Hall/CRC Boca Raton, (1999). Google Scholar

[29]

J. C. Maxwell, Theory of Heat,, Cambridge University Press, (2011). doi: 10.1017/CBO9781139057943. Google Scholar

[30]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of memory type in thermoelasticity of type III,, Appl. Anal., 87 (2008), 13. doi: 10.1080/00036810701714180. Google Scholar

[31]

S. Mukhopadyay, R. Prasad and R. Kumar, On the theory of two-temeperature thermoelasticity with two phase-lags,, J. Thermal Stresses, 34 (2011), 352. doi: 10.1080/01495739.2010.550815. Google Scholar

[32]

P. Puri and P. M. Jordan, On the propagation of plane waves in type-III thermoelastic media,, Proc. Roy. Soc. London A, 460 (2004), 3203. doi: 10.1098/rspa.2004.1341. Google Scholar

[33]

Y. Qin, S. Deng, L. Huang, Z. Ma and X. Su, Global existence for the three-dimensional thermoelastic equations of Type II,, Quart. Appl. Math., 68 (2010), 333. Google Scholar

[34]

R. Quintanilla, On the spatial behaviour in thermoelasticity without energy dissipation,, J. Thermal Stresses, 22 (1999), 213. doi: 10.1080/014957399280977. Google Scholar

[35]

R. Quintanilla, Damping of end effects in a thermoelastic theory,, Appl. Math. Lett., 14 (2001), 137. doi: 10.1016/S0893-9659(00)00125-7. Google Scholar

[36]

R. Quintanilla, Convergence and structural stability in thermoelasticity,, Appl. Math. Comput., 135 (2003), 287. doi: 10.1016/S0096-3003(01)00331-9. Google Scholar

[37]

R. Quintanilla, Exponential stability and uniqueness in thermoelasticity with two temperatures,, Dyn. Conti. Discrete Impuls. Syst. Ser. A: Math. Anal., 11 (2004), 57. Google Scholar

[38]

R. Quintanilla, On the impossibility of localization in linear thermoelasticity,, Proc. Roy. Soc. London A, 463 (2007), 3311. doi: 10.1098/rspa.2007.0076. Google Scholar

[39]

R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction,, J. Thermal Stresses, 31 (2008), 260. doi: 10.1080/01495730701738272. Google Scholar

[40]

R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction,, J. Thermal Stresses, 32 (2009), 1270. doi: 10.1080/01495730903310599. Google Scholar

[41]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type III,, Discrete Contin. Dyn. Syst., 3 (2003), 383. doi: 10.3934/dcdsb.2003.3.383. Google Scholar

[42]

R. Quintanilla and G. Saccomandi, Phragmén-Lindelöf alternative of exponential type for the solutions of a fourth order dispersive equation,, Rend. Lincei Mat. Appl., 23 (2012), 105. doi: 10.4171/RLM/620. Google Scholar

[43]

R. Quintanilla and B. Straughan, Growth and uniqueness in thermoelasticity,, Proc. Roy. Soc. London A, 456 (2000), 1419. doi: 10.1098/rspa.2000.0569. Google Scholar

[44]

R. Quintanilla and B. Straughan, Energy bounds for some non-standard problems in thermoelasticity,, Proc. Roy. Soc. London A, 461 (2005), 1147. doi: 10.1098/rspa.2004.1381. Google Scholar

[45]

S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model,, J. Thermal Stresses, 30 (2007), 231. doi: 10.1080/01495730601130919. Google Scholar

[46]

B. Straughan, Heat Waves,, Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4614-0493-4. Google Scholar

[47]

D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales,, ASME J. Heat Transfer, 117 (1995), 8. doi: 10.1115/1.2822329. Google Scholar

[48]

W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelasticity,, Acta Mechanica, 16 (1973), 21. doi: 10.1007/BF01177123. Google Scholar

[49]

L. Yang and Y. G. Wang, Well-posedness and decay estimates for Cauchy problems of linear thermoelastic systems of type III in 3-D,, Indiana Univ. Math. J., 55 (2006), 1333. doi: 10.1512/iumj.2006.55.2799. Google Scholar

[50]

H. M. Youssef, Theory of two-temperature thermoelasticity without energy dissipation,, J. Thermal Stresses, 34 (2011), 138. doi: 10.1080/01495739.2010.511941. Google Scholar

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