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

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  • 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.
    Mathematics Subject Classification: Primary: 35Q79; Secondary: 35B40, 35B35, 80A20.

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  • [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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