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

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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

M. Carme Leseduarte ^1, and Ramon Quintanilla ^1,

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

Abstract
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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.

Keywords: existence, Backward in time problem, spatial growth/decay of solutions, Type III with two temperatures thermoelasticity, energy arguments., continuous dependence.

Mathematics Subject Classification: Primary: 35Q79; Secondary: 35B53, 35B35, 74F0.

Citation: M. Carme Leseduarte, Ramon Quintanilla. On the backward in time problem for the thermoelasticity with two temperatures. Discrete and 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), 147-160. doi: 10.1080/01495739.2010.511942.
[2]	P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, Jour. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627. doi: 10.1007/BF01594969.
[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), 969-970. doi: 10.1007/BF01602278.
[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), 107-112. doi: 10.1007/BF01591120.
[5]	J. I. Díaz and R. Quintanilla, Spatial and contiuous dependence estimates in linear viscoelastity, J. Math. Anal. Appl., 273 (2002), 1-16. doi: 10.1016/S0022-247X(02)00200-7.
[6]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Lett., 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010.
[7]	A. S. El-Karamany and M. A. Ezzat, On the two-temperature Green-Naghdi thermoelasticity theories, J. Thermal Stresses, 34 (2011), 1207-1226. doi: 10.1080/01495739.2011.608313.
[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), 325-350.
[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), Chichester: Ellis Horwood, (1989) pp. 101-111.
[10]	J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford: Oxford Mathematical Monographs, Oxford, 1985.
[11]	A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.
[12]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.
[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), 379-388.
[14]	R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-476. doi: 10.1080/014957399280832.
[15]	R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, Int. J. Solids Struct., 37 (1999), 215-224. doi: 10.1016/S0020-7683(99)00089-X.
[16]	C. O. Horgan, L. E. Payne and L. T. Wheeler, Spatial decay estimates in transient heat conduction, Quart. Appl. Math., 42 (1984), 119-127.
[17]	C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects in functionally graded heat conducting materials, Quart. Appl. Math., 59 (2001), 529-542.
[18]	D. Iesan, On the theory of thermoelasticity without energy dissipation, J. Thermal Stresses, 21 (1998), 295-307. doi: 10.1080/01495739808956148.
[19]	D. Iesan, Thermopiezoelectricity without energy dissipation, Proc. Roy. Soc. London A, 464 (2008), 631-656. doi: 10.1098/rspa.2007.0264.
[20]	D. Iesan and R. Quintanilla, On the thermoelastic bodies with inner structure and microtemperatures, J. Math. Anal. Appl., 354 (2009), 12-23. doi: 10.1016/j.jmaa.2008.12.017.
[21]	J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford: Oxford Mathematical Monographs, Oxford, 2010.
[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), 317-329. doi: 10.1016/j.jmaa.2007.05.017.
[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., Ser. B, 13 (2010), 375-391. doi: 10.3934/dcdsb.2010.13.375.
[24]	M. C. Leseduarte and R. Quintanilla, On uniqueness and continuous dependence in type III thermoelasticity, J. Math. Anal. Appl., 395 (2012), 429-436. doi: 10.1016/j.jmaa.2012.05.019.
[25]	M. C. Leseduarte and R. Quintanilla, Phragmén-Lindelöf alternative for an exact heat conduction equation with delay, Commun. Pure Appl. Anal., 12 (2013), 1221-1235. doi: 10.3934/cpaa.2013.12.1221.
[26]	Y. Liu and C. Lin, Phragmén-Lindelöf alternative and continuous dependence-type results for the thermoelasticity of type III, Appl. Anal., 87 (2008), 431-449. doi: 10.1080/00036810801927963.
[27]	Z. Liu and R. Quintanilla, Energy decay rates of mixed type II and type III thermoelastic system, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 1433-1444. doi: 10.3934/dcdsb.2010.14.1433.
[28]	Z. Liu and S. Zheng, Semigroups Associated to Dissipative Systems, Chapman & Hall/CRC Boca Raton, FL. Research Notes in Mathematics, vol. 398, 1999.
[29]	J. C. Maxwell, Theory of Heat, Cambridge University Press, Dover, Mineola, New York, 2011. doi: 10.1017/CBO9781139057943.
[30]	S. A. Messaoudi and A. Soufyane, Boundary stabilization of memory type in thermoelasticity of type III, Appl. Anal., 87 (2008), 13-28. doi: 10.1080/00036810701714180.
[31]	S. Mukhopadyay, R. Prasad and R. Kumar, On the theory of two-temeperature thermoelasticity with two phase-lags, J. Thermal Stresses, 34 (2011), 352-365. doi: 10.1080/01495739.2010.550815.
[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), 3203-3221. doi: 10.1098/rspa.2004.1341.
[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), 333-348.
[34]	R. Quintanilla, On the spatial behaviour in thermoelasticity without energy dissipation, J. Thermal Stresses, 22 (1999), 213-224. doi: 10.1080/014957399280977.
[35]	R. Quintanilla, Damping of end effects in a thermoelastic theory, Appl. Math. Lett., 14 (2001), 137-141. doi: 10.1016/S0893-9659(00)00125-7.
[36]	R. Quintanilla, Convergence and structural stability in thermoelasticity, Appl. Math. Comput., 135 (2003), 287-300. doi: 10.1016/S0096-3003(01)00331-9.
[37]	R. Quintanilla, Exponential stability and uniqueness in thermoelasticity with two temperatures, Dyn. Conti. Discrete Impuls. Syst. Ser. A: Math. Anal., 11 (2004), 57-68.
[38]	R. Quintanilla, On the impossibility of localization in linear thermoelasticity, Proc. Roy. Soc. London A, 463 (2007), 3311-3322. doi: 10.1098/rspa.2007.0076.
[39]	R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269. doi: 10.1080/01495730701738272.
[40]	R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599.
[41]	R. Quintanilla and R. Racke, Stability in thermoelasticity of type III, Discrete Contin. Dyn. Syst., Ser. B, 3 (2003), 383-400. doi: 10.3934/dcdsb.2003.3.383.
[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), 105-113. doi: 10.4171/RLM/620.
[43]	R. Quintanilla and B. Straughan, Growth and uniqueness in thermoelasticity, Proc. Roy. Soc. London A, 456 (2000), 1419-1429. doi: 10.1098/rspa.2000.0569.
[44]	R. Quintanilla and B. Straughan, Energy bounds for some non-standard problems in thermoelasticity, Proc. Roy. Soc. London A, 461 (2005), 1147-1162. doi: 10.1098/rspa.2004.1381.
[45]	S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919.
[46]	B. Straughan, Heat Waves, Applied Mathematical Sciences, 177. Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4.
[47]	D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), pp. 8-16. doi: 10.1115/1.2822329.
[48]	W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelasticity, Acta Mechanica, 16 (1973), 21-33. doi: 10.1007/BF01177123.
[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), 1333-1361. doi: 10.1512/iumj.2006.55.2799.
[50]	H. M. Youssef, Theory of two-temperature thermoelasticity without energy dissipation, J. Thermal Stresses, 34 (2011), 138-146. doi: 10.1080/01495739.2010.511941.

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), 147-160. doi: 10.1080/01495739.2010.511942.
[2]	P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, Jour. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627. doi: 10.1007/BF01594969.
[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), 969-970. doi: 10.1007/BF01602278.
[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), 107-112. doi: 10.1007/BF01591120.
[5]	J. I. Díaz and R. Quintanilla, Spatial and contiuous dependence estimates in linear viscoelastity, J. Math. Anal. Appl., 273 (2002), 1-16. doi: 10.1016/S0022-247X(02)00200-7.
[6]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Lett., 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010.
[7]	A. S. El-Karamany and M. A. Ezzat, On the two-temperature Green-Naghdi thermoelasticity theories, J. Thermal Stresses, 34 (2011), 1207-1226. doi: 10.1080/01495739.2011.608313.
[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), 325-350.
[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), Chichester: Ellis Horwood, (1989) pp. 101-111.
[10]	J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford: Oxford Mathematical Monographs, Oxford, 1985.
[11]	A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.
[12]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.
[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), 379-388.
[14]	R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-476. doi: 10.1080/014957399280832.
[15]	R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, Int. J. Solids Struct., 37 (1999), 215-224. doi: 10.1016/S0020-7683(99)00089-X.
[16]	C. O. Horgan, L. E. Payne and L. T. Wheeler, Spatial decay estimates in transient heat conduction, Quart. Appl. Math., 42 (1984), 119-127.
[17]	C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects in functionally graded heat conducting materials, Quart. Appl. Math., 59 (2001), 529-542.
[18]	D. Iesan, On the theory of thermoelasticity without energy dissipation, J. Thermal Stresses, 21 (1998), 295-307. doi: 10.1080/01495739808956148.
[19]	D. Iesan, Thermopiezoelectricity without energy dissipation, Proc. Roy. Soc. London A, 464 (2008), 631-656. doi: 10.1098/rspa.2007.0264.
[20]	D. Iesan and R. Quintanilla, On the thermoelastic bodies with inner structure and microtemperatures, J. Math. Anal. Appl., 354 (2009), 12-23. doi: 10.1016/j.jmaa.2008.12.017.
[21]	J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford: Oxford Mathematical Monographs, Oxford, 2010.
[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), 317-329. doi: 10.1016/j.jmaa.2007.05.017.
[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., Ser. B, 13 (2010), 375-391. doi: 10.3934/dcdsb.2010.13.375.
[24]	M. C. Leseduarte and R. Quintanilla, On uniqueness and continuous dependence in type III thermoelasticity, J. Math. Anal. Appl., 395 (2012), 429-436. doi: 10.1016/j.jmaa.2012.05.019.
[25]	M. C. Leseduarte and R. Quintanilla, Phragmén-Lindelöf alternative for an exact heat conduction equation with delay, Commun. Pure Appl. Anal., 12 (2013), 1221-1235. doi: 10.3934/cpaa.2013.12.1221.
[26]	Y. Liu and C. Lin, Phragmén-Lindelöf alternative and continuous dependence-type results for the thermoelasticity of type III, Appl. Anal., 87 (2008), 431-449. doi: 10.1080/00036810801927963.
[27]	Z. Liu and R. Quintanilla, Energy decay rates of mixed type II and type III thermoelastic system, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 1433-1444. doi: 10.3934/dcdsb.2010.14.1433.
[28]	Z. Liu and S. Zheng, Semigroups Associated to Dissipative Systems, Chapman & Hall/CRC Boca Raton, FL. Research Notes in Mathematics, vol. 398, 1999.
[29]	J. C. Maxwell, Theory of Heat, Cambridge University Press, Dover, Mineola, New York, 2011. doi: 10.1017/CBO9781139057943.
[30]	S. A. Messaoudi and A. Soufyane, Boundary stabilization of memory type in thermoelasticity of type III, Appl. Anal., 87 (2008), 13-28. doi: 10.1080/00036810701714180.
[31]	S. Mukhopadyay, R. Prasad and R. Kumar, On the theory of two-temeperature thermoelasticity with two phase-lags, J. Thermal Stresses, 34 (2011), 352-365. doi: 10.1080/01495739.2010.550815.
[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), 3203-3221. doi: 10.1098/rspa.2004.1341.
[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), 333-348.
[34]	R. Quintanilla, On the spatial behaviour in thermoelasticity without energy dissipation, J. Thermal Stresses, 22 (1999), 213-224. doi: 10.1080/014957399280977.
[35]	R. Quintanilla, Damping of end effects in a thermoelastic theory, Appl. Math. Lett., 14 (2001), 137-141. doi: 10.1016/S0893-9659(00)00125-7.
[36]	R. Quintanilla, Convergence and structural stability in thermoelasticity, Appl. Math. Comput., 135 (2003), 287-300. doi: 10.1016/S0096-3003(01)00331-9.
[37]	R. Quintanilla, Exponential stability and uniqueness in thermoelasticity with two temperatures, Dyn. Conti. Discrete Impuls. Syst. Ser. A: Math. Anal., 11 (2004), 57-68.
[38]	R. Quintanilla, On the impossibility of localization in linear thermoelasticity, Proc. Roy. Soc. London A, 463 (2007), 3311-3322. doi: 10.1098/rspa.2007.0076.
[39]	R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269. doi: 10.1080/01495730701738272.
[40]	R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599.
[41]	R. Quintanilla and R. Racke, Stability in thermoelasticity of type III, Discrete Contin. Dyn. Syst., Ser. B, 3 (2003), 383-400. doi: 10.3934/dcdsb.2003.3.383.
[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), 105-113. doi: 10.4171/RLM/620.
[43]	R. Quintanilla and B. Straughan, Growth and uniqueness in thermoelasticity, Proc. Roy. Soc. London A, 456 (2000), 1419-1429. doi: 10.1098/rspa.2000.0569.
[44]	R. Quintanilla and B. Straughan, Energy bounds for some non-standard problems in thermoelasticity, Proc. Roy. Soc. London A, 461 (2005), 1147-1162. doi: 10.1098/rspa.2004.1381.
[45]	S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919.
[46]	B. Straughan, Heat Waves, Applied Mathematical Sciences, 177. Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4.
[47]	D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), pp. 8-16. doi: 10.1115/1.2822329.
[48]	W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelasticity, Acta Mechanica, 16 (1973), 21-33. doi: 10.1007/BF01177123.
[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), 1333-1361. doi: 10.1512/iumj.2006.55.2799.
[50]	H. M. Youssef, Theory of two-temperature thermoelasticity without energy dissipation, J. Thermal Stresses, 34 (2011), 138-146. doi: 10.1080/01495739.2010.511941.

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