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May  2013, 12(3): 1221-1235. doi: 10.3934/cpaa.2013.12.1221

## Phragmén-Lindelöf alternative for an exact heat conduction equation with delay

 1 Departament de Matemàtica Aplicada 2, ETSEIAT–UPC, C. Colom 11, 08222 Terrassa, Barcelona, Spain 2 Matemática Aplicada 2, E.T.S.E.I.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain

Received  November 2011 Revised  May 2012 Published  September 2012

In this paper we investigate the spatial behavior of the solutions for a theory for the heat conduction with one delay term. We obtain a Phragmén-Lindelöf type alternative. That is, the solutions either decay in an exponential way or blow-up at infinity in an exponential way. We also show how to obtain an upper bound for the amplitude term. Later we point out how to extend the results to a thermoelastic problem. We finish the paper by considering the equation obtained by the Taylor approximation to the delay term. A Phragmén-Lindelöf type alternative is obtained for the forward and backward in time equations.
Citation: M. Carme Leseduarte, Ramon Quintanilla. Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1221-1235. doi: 10.3934/cpaa.2013.12.1221
##### References:
 [1] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729. doi: 10.1115/1.3098984. [2] M. Dreher, R. Quintanilla and R. Racke, Ill posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010. [3] J. N. Flavin, R. J. Knops and L. E. Payne, Decay estimates for the constrained elastic cylinder of variable cross-section, Quarterly Applied Mathematics, 47 (1989), 325-350. [4] 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'' (G. Eason and R.W. Ogden eds.), Chichester: Ellis Horwood, (1989), pp. 101-111. [5] R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470. doi: 10.1080/014957399280832. [6] R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, International Journal of Solids and Structures, 37 (2000), 215-224. doi: 10.1016/S0020-7683(99)00089-X. [7] C. O. Horgan, L. E. Payne and L. T. Wheeler, Spatial decay estimates in transient heat conduction, Quarterly Applied Mathematics, 42 (1984), 119-127. [8] C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects in functionally graded heat conducting materials, Quarterly Applied Mathematics, 59 (2001), 529-542. [9] C. O. Horgan and R. Quintanilla, Spatial behaviour of solutions of the dual-phase-lag heat equation, Math. Methods Appl. Sci., 28 (2005), 43-57. doi: 10.1002/mma.548. [10] J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticity with Finite Wave Speeds,'' Oxford Mathematical Monographs, Oxford, 2010. [11] R. Kumar and S. Mukhopadhyay, Analysis of the effects of phase-lags on propagation of harmonic plane waves in thermoelastic media, Comp. Methods in Sci. Tech., 16 (2010), 19-28. [12] M. C. Leseduarte and R. Quintanilla, Some qualitative properties of solutions of the system governing acoustic waves in bubbly liquids, International Journal of Engineering Science, 44 (2006), 1146-1155. doi: 10.1016/j.ijengsci.2006.06.009. [13] M. C. Leseduarte and R. Quintanilla, Spatial behavior for solutions in heat conduction with two delays, Manuscript, (2011). [14] S. Mukhopadhyay and R. Kumar, Analysis of phase-lag effects on wave propagation in a thick plate under axisymmetric temperature distribution, Acta Mechanica, 210 (2010), 331-344. doi: 10.1007/s00707-009-0209-9. [15] S. Mukhopadhyay, S. Kothari and R. Kumar, On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags, Acta Mechanica, 214 (2010), 305-314. doi: 10.1007/s00707-010-0291-z. [16] R. Quintanilla, Damping of end effects in a thermoelastic theory, Appl. Math. Letters, 14 (2001), 137-141. doi: 10.1016/S0893-9659(00)00125-7. [17] R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilibrium Thermodynamics, 27 (2002), 217-227. doi: 10.1515/JNETDY.2002.012. [18] R. Quintanilla, A condition on the delay parameters in the one-dimensional dual-phase-lag thermoelastic theory, J. Thermal Stresses, 26 (2003), 713-721. doi: 10.1080/713855996. [19] R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, Journal of Thermal Stresses, 31 (2008), 260-269. doi: 10.1080/01495730701738272. [20] R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, Journal of Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599. [21] R. Quintanilla, Spatial estimates for an equation with a delay term, Journal Applied Mathematics Physics (ZAMP), 61 (2010), 381-388. doi: 10.1007/s00033-009-0049-4. [22] R. Quintanilla, Some solutions for a family of exact phase-lag heat conduction problems, Mechanics Research Communications, 38 (2011), 355-360. doi: 10.1016/j.mechrescom.2011.04.008. [23] R. Quintanilla and R. Racke, A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213. doi: 10.1016/j.ijheatmasstransfer.2005.10.016. [24] R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM Journal of Applied Mathematics, 66 (2006), 977-1001. doi: 10.1137/05062860X. [25] R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A, 463 (2007), 659-674. doi: 10.1098/rspa.2006.1784. [26] R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29. doi: 10.1016/j.ijheatmasstransfer.2007.04.045. [27] S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919. [28] B. Straughan, "Heat Waves,'' Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-1-4614-0493-4. [29] D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16. doi: 10.1115/1.2822329. [30] L. Wang, X. Zhou and X. Wei, "Heat Conduction, Mathematical Models and Analytical Solutions,'' Springer-Verlag, Berlin Heidelberg, 2008. [31] F. Xu, S. Moon, X. Zhang, L. Shao, Y. S. Song and U. Demirci, Multi-scale heat and mass transfer modelling of cell and tissue cryopreservation, Phyl. Transactions Royal Society A-Math. Phys. and Engin. Scies., 368 (2010), 561-583. doi: 10.1098/rsta.2009.0248. [32] F. Xu, T. J. Lu and X. E. Guo, Multi-scale biothermal and biomechanical behaviours of biological materials, Phyl. Transactions Royal Society A-Math. Phys. and Engin. Scies., 368 (2010), 517-519. doi: 10.1098/rsta.2009.0249.

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##### References:
 [1] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729. doi: 10.1115/1.3098984. [2] M. Dreher, R. Quintanilla and R. Racke, Ill posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010. [3] J. N. Flavin, R. J. Knops and L. E. Payne, Decay estimates for the constrained elastic cylinder of variable cross-section, Quarterly Applied Mathematics, 47 (1989), 325-350. [4] 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'' (G. Eason and R.W. Ogden eds.), Chichester: Ellis Horwood, (1989), pp. 101-111. [5] R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470. doi: 10.1080/014957399280832. [6] R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, International Journal of Solids and Structures, 37 (2000), 215-224. doi: 10.1016/S0020-7683(99)00089-X. [7] C. O. Horgan, L. E. Payne and L. T. Wheeler, Spatial decay estimates in transient heat conduction, Quarterly Applied Mathematics, 42 (1984), 119-127. [8] C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects in functionally graded heat conducting materials, Quarterly Applied Mathematics, 59 (2001), 529-542. [9] C. O. Horgan and R. Quintanilla, Spatial behaviour of solutions of the dual-phase-lag heat equation, Math. Methods Appl. Sci., 28 (2005), 43-57. doi: 10.1002/mma.548. [10] J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticity with Finite Wave Speeds,'' Oxford Mathematical Monographs, Oxford, 2010. [11] R. Kumar and S. Mukhopadhyay, Analysis of the effects of phase-lags on propagation of harmonic plane waves in thermoelastic media, Comp. Methods in Sci. Tech., 16 (2010), 19-28. [12] M. C. Leseduarte and R. Quintanilla, Some qualitative properties of solutions of the system governing acoustic waves in bubbly liquids, International Journal of Engineering Science, 44 (2006), 1146-1155. doi: 10.1016/j.ijengsci.2006.06.009. [13] M. C. Leseduarte and R. Quintanilla, Spatial behavior for solutions in heat conduction with two delays, Manuscript, (2011). [14] S. Mukhopadhyay and R. Kumar, Analysis of phase-lag effects on wave propagation in a thick plate under axisymmetric temperature distribution, Acta Mechanica, 210 (2010), 331-344. doi: 10.1007/s00707-009-0209-9. [15] S. Mukhopadhyay, S. Kothari and R. Kumar, On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags, Acta Mechanica, 214 (2010), 305-314. doi: 10.1007/s00707-010-0291-z. [16] R. Quintanilla, Damping of end effects in a thermoelastic theory, Appl. Math. Letters, 14 (2001), 137-141. doi: 10.1016/S0893-9659(00)00125-7. [17] R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilibrium Thermodynamics, 27 (2002), 217-227. doi: 10.1515/JNETDY.2002.012. [18] R. Quintanilla, A condition on the delay parameters in the one-dimensional dual-phase-lag thermoelastic theory, J. Thermal Stresses, 26 (2003), 713-721. doi: 10.1080/713855996. [19] R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, Journal of Thermal Stresses, 31 (2008), 260-269. doi: 10.1080/01495730701738272. [20] R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, Journal of Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599. [21] R. Quintanilla, Spatial estimates for an equation with a delay term, Journal Applied Mathematics Physics (ZAMP), 61 (2010), 381-388. doi: 10.1007/s00033-009-0049-4. [22] R. Quintanilla, Some solutions for a family of exact phase-lag heat conduction problems, Mechanics Research Communications, 38 (2011), 355-360. doi: 10.1016/j.mechrescom.2011.04.008. [23] R. Quintanilla and R. Racke, A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213. doi: 10.1016/j.ijheatmasstransfer.2005.10.016. [24] R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM Journal of Applied Mathematics, 66 (2006), 977-1001. doi: 10.1137/05062860X. [25] R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A, 463 (2007), 659-674. doi: 10.1098/rspa.2006.1784. [26] R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29. doi: 10.1016/j.ijheatmasstransfer.2007.04.045. [27] S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919. [28] B. Straughan, "Heat Waves,'' Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-1-4614-0493-4. [29] D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16. doi: 10.1115/1.2822329. [30] L. Wang, X. Zhou and X. Wei, "Heat Conduction, Mathematical Models and Analytical Solutions,'' Springer-Verlag, Berlin Heidelberg, 2008. [31] F. Xu, S. Moon, X. Zhang, L. Shao, Y. S. Song and U. Demirci, Multi-scale heat and mass transfer modelling of cell and tissue cryopreservation, Phyl. Transactions Royal Society A-Math. Phys. and Engin. Scies., 368 (2010), 561-583. doi: 10.1098/rsta.2009.0248. [32] F. Xu, T. J. Lu and X. E. Guo, Multi-scale biothermal and biomechanical behaviours of biological materials, Phyl. Transactions Royal Society A-Math. Phys. and Engin. Scies., 368 (2010), 517-519. doi: 10.1098/rsta.2009.0249.
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