• Previous Article
    Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data
  • CPAA Home
  • This Issue
  • Next Article
    On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent
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 & 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. 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. 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. 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, (1989), 101. Google Scholar

[5]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity,, J. Thermal Stresses, 22 (1999), 451. 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. 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. 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. 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. doi: 10.1002/mma.548. Google Scholar

[10]

J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticity with Finite Wave Speeds,'', Oxford Mathematical Monographs, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1080/01495730601130919. Google Scholar

[28]

B. Straughan, "Heat Waves,'', Springer-Verlag, (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. doi: 10.1115/1.2822329. Google Scholar

[30]

L. Wang, X. Zhou and X. Wei, "Heat Conduction, Mathematical Models and Analytical Solutions,'', Springer-Verlag, (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. 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. 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. 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. 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. 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, (1989), 101. Google Scholar

[5]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity,, J. Thermal Stresses, 22 (1999), 451. 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. 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. 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. 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. doi: 10.1002/mma.548. Google Scholar

[10]

J. Ignaczak and M. Ostoja-Starzewski, "Thermoelasticity with Finite Wave Speeds,'', Oxford Mathematical Monographs, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1080/01495730601130919. Google Scholar

[28]

B. Straughan, "Heat Waves,'', Springer-Verlag, (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. doi: 10.1115/1.2822329. Google Scholar

[30]

L. Wang, X. Zhou and X. Wei, "Heat Conduction, Mathematical Models and Analytical Solutions,'', Springer-Verlag, (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. 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. doi: 10.1098/rsta.2009.0249. Google Scholar

[1]

Fabio Punzo. Phragmèn-Lindelöf principles for fully nonlinear elliptic equations with unbounded coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1439-1461. doi: 10.3934/cpaa.2010.9.1439

[2]

Seppo Granlund, Niko Marola. Phragmén--Lindelöf theorem for infinity harmonic functions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 127-132. doi: 10.3934/cpaa.2015.14.127

[3]

Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165

[4]

Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. Heat conduction with memory: A singular kernel problem. Evolution Equations & Control Theory, 2014, 3 (3) : 399-410. doi: 10.3934/eect.2014.3.399

[5]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014

[6]

Kazuhiro Ishige, Tatsuki Kawakami. Asymptotic behavior of solutions for some semilinear heat equations in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1351-1371. doi: 10.3934/cpaa.2009.8.1351

[7]

Cristina Brändle, Arturo De Pablo. Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1161-1178. doi: 10.3934/cpaa.2018056

[8]

Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793

[9]

Juan Campos, Rafael Obaya, Massimo Tarallo. Recurrent equations with sign and Fredholm alternative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 959-977. doi: 10.3934/dcdss.2016036

[10]

Corrado Mascia. Stability analysis for linear heat conduction with memory kernels described by Gamma functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3569-3584. doi: 10.3934/dcds.2015.35.3569

[11]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078

[12]

Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133

[13]

Xianyi Li, Deming Zhu. Comparison theorems of oscillation and nonoscillation for neutral difference equations with continuous arguments. Communications on Pure & Applied Analysis, 2003, 2 (4) : 579-589. doi: 10.3934/cpaa.2003.2.579

[14]

Alina Gleska, Małgorzata Migda. Qualitative properties of solutions of higher order difference equations with deviating arguments. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 239-252. doi: 10.3934/dcdsb.2018016

[15]

Delio Mugnolo. Gaussian estimates for a heat equation on a network. Networks & Heterogeneous Media, 2007, 2 (1) : 55-79. doi: 10.3934/nhm.2007.2.55

[16]

Norisuke Ioku. Some space-time integrability estimates of the solution for heat equations in two dimensions. Conference Publications, 2011, 2011 (Special) : 707-716. doi: 10.3934/proc.2011.2011.707

[17]

Olof Heden, Faina I. Solov’eva. Partitions of $\mathbb F$n into non-parallel Hamming codes. Advances in Mathematics of Communications, 2009, 3 (4) : 385-397. doi: 10.3934/amc.2009.3.385

[18]

Xavier Cabré, Eleonora Cinti. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1179-1206. doi: 10.3934/dcds.2010.28.1179

[19]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45

[20]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1105-1132. doi: 10.3934/dcds.2003.9.1105

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]