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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

M. Carme Leseduarte 1, and  Ramon Quintanilla 2,

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.
Keywords: equations with delay, Spatial estimates, energy arguments, heat conduction., Phragmén-Lindelöf alternative.
Mathematics Subject Classification: Primary: 35Q79; Secondary: 35B40, 35B35, 80A2.
Citation: M. Carme Leseduarte, Ramon Quintanilla. Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470. doi: 10.1080/014957399280832.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[13]

M. C. Leseduarte and R. Quintanilla, Spatial behavior for solutions in heat conduction with two delays, Manuscript, (2011). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919.  Google Scholar

[28]

B. Straughan, "Heat Waves,'' Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar

[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.  Google Scholar

[30]

L. Wang, X. Zhou and X. Wei, "Heat Conduction, Mathematical Models and Analytical Solutions,'' Springer-Verlag, Berlin Heidelberg, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729. doi: 10.1115/1.3098984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470. doi: 10.1080/014957399280832.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[13]

M. C. Leseduarte and R. Quintanilla, Spatial behavior for solutions in heat conduction with two delays, Manuscript, (2011). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. K. Roy Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919.  Google Scholar

[28]

B. Straughan, "Heat Waves,'' Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar

[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.  Google Scholar

[30]

L. Wang, X. Zhou and X. Wei, "Heat Conduction, Mathematical Models and Analytical Solutions,'' Springer-Verlag, Berlin Heidelberg, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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