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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 |
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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