# American Institute of Mathematical Sciences

March  2019, 8(1): 31-42. doi: 10.3934/eect.2019002

## Some remarks on the model of rigid heat conductor with memory: Unbounded heat relaxation function

 1 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma La Sapienza, Via A. Scarpa 16, I-00161, Italy 2 I.N.F.N. - Sez. Roma1, Gr.Ⅳ: Mathematical Methods in NonLinear Physics, Rome, Italy

Received  March 2018 Revised  July 2018 Published  March 2019 Early access  January 2019

Fund Project: The author is supported by G.N.F.M.-I.N.d.A.M., I.N.F.N. and La Sapienza Università di Roma, Italy.

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function $k$ is assumed to be unbounded at the initial time $t = 0$. That is, it is represented by a regular integrable function, namely $k\in L^1( \mathbb{R}^+)$, but its time derivative is not integrable, that is $\dot k\notin L^1( \mathbb{R}^+)$. The study takes its origin in [2]: the heat conductor model described therein is modified in such a way to adapt it to the case of a heat flux relaxation function $k$ which is unbounded at $t = 0$. Notably, also when these relaxed assumptions on $k$ are introduced, whenever two different thermal states which correspond to the same heat flux are considered, then both states correspond also to the same thermal work. Accordingly, the notion of equivalence can be introduced, together with its physical relevance, both in the regular kernel case in [2] as well as in the singular kernel case analysed in the present investigation.

Citation: Sandra Carillo. Some remarks on the model of rigid heat conductor with memory: Unbounded heat relaxation function. Evolution Equations and Control Theory, 2019, 8 (1) : 31-42. doi: 10.3934/eect.2019002
##### References:
 [1] M. J. Ablowitz and A.S. Fokas, Complex Variables, Cambridge University Press, Cambridge, 1997. [2] G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. of Mech. and Appl. Math., 57 (2004), 429-446.  doi: 10.1093/qjmam/57.3.429. [3] L. Boltzmann, Zur theorie der elastichen nachwirkung, Annalen der physik und chemie, 77 (1876), 624-654. [4] S. Carillo, Regular and singular kernel problems in magneto-viscoelasticity, Meccanica S.I. "New trends in Dynamics and Stability", 52 (2017), 3053-3060.  doi: 10.1007/s11012-017-0722-1. [5] S. Carillo, Existence, uniqueness and exponential decay: An evolution problem in heat conduction with memory, Quart. Appl. Math., 69 (2011), 635-649.  doi: 10.1090/S0033-569X-2011-01223-1. [6] S. Carillo, An evolution problem in materials with fading memory: Solution's existence and uniqueness, Complex Var. Elliptic Equ., 56 (2011), 481-492.  doi: 10.1080/17476931003786667. [7] S. Carillo, Materials with memory: Free energies & solution exponential decay, Commun. Pure Appl. Anal., 9 (2010), 1235-1248.  doi: 10.3934/cpaa.2010.9.1235. [8] S. Carillo, A 3-Dimensional Singular Kernel Problem in Viscoelasticity: An Existence Result, Atti Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali, ISSN 1825-1242 96, No. S2, A1 (2018), arXiv:1808.02411. doi: 10.1478/AAPP.96S2A1. [9] S. Carillo and C. Giorgi, Non-classical Memory Kernels in Linear Viscoelasticity, Chapter 13 in "Viscoelastic and Viscoplastic Materials", M.F. El-Amin Editor, ISBN 978-953-51-2603-4, Print ISBN 978-953-51-2602-7, Published: September 6, 2016, InTech. doi: 10.5772/64251. [10] S. Carillo, Singular kernel problems in materials with memory, Meccanica, 50 (2015), 603-615.  doi: 10.1007/s11012-014-0083-y. [11] S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, A magneto-viscoelasticity problem with a singular memory kernel, Nonlinear Analysis Series B: Real World Applications, 35 (2017), 200-210.  doi: 10.1016/j.nonrwa.2016.10.014. [12] S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, On weak regularity requirements of the relaxation modulus in viscoelasticity, arXiv: 1811.06723, (2018). [13] S. Carillo, V. Valente and G. Vergara Caffarelli, A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity, Applicable Analisys: An International Journal, 1563-504X, 90 (2011), 1791-1802. doi: 10.1080/00036811003735832. [14] S. Carillo, V. Valente and G. Vergara Caffarelli, An existence theorem for the magnetic-viscoelastic problem, Discrete and Continuous Dynamical Systems Series S., 5 (2012), 435-447.  doi: 10.3934/dcdss.2012.5.435. [15] S. Carillo, V. Valente and G. Vergara Caffarelli, Heat conduction with memory: A singular kernel problem, Evolution Equations and Control Theory, 3 (2014), 399-410.  doi: 10.3934/eect.2014.3.399. [16] S. Carillo, V. Valente and G. Vergara Caffarelli, A linear viscoelasticity problem with a singular memory kernel: An existence and uniqueness result, Differential and Integral Equations, 26 (2013), 1115-1125. [17] S. Carillo, Some remarks on materials with memory: Heat conduction and viscoelasticity, Journal of Nonlinear Mathematical Physics, 12 (2005), 163-178.  doi: 10.2991/jnmp.2005.12.s1.14. [18] S. Carillo, Some remarks on materials with memory: Heat conduction and viscoelasticity, Journal of Nonlinear Mathematical Physics, 3 (2015), i-iii.  doi: 10.1080/14029251.2014.971573. [19] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Universitá Modena, 3 (1949), 83-101. [20] J. Ciambella, A. Paolone and S. Vidoli, Memory decay rates of viscoelastic solids: Not too slow, but not too fast either, Rheol. Acta, 50 (2011), 661-674.  doi: 10.1007/s00397-011-0549-y. [21] B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal., 17 (1964), 1-46.  doi: 10.1007/BF00283864. [22] B. D. Coleman and E. H. Dill, On thermodynamics and stability of materials with memory, Arch. Rat. Mech. Anal., 51 (1973), 1-53.  doi: 10.1007/BF00275991. [23] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Diff. Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4. [24] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [25] B. de Andrade and A. Viana, On a fractional reaction diffusion equation, Z. Angew. Math. Phys., 68 (2017), Art. 59, 11 pp. doi: 10.1007/s00033-017-0801-0. [26] L. Deseri, M. Zingales and P. Pollaci, The state of fractional hereditary materials (FHM), Discrete and Continuous Dynamical Systems - B, 19 (2014), 2065-2089.  doi: 10.3934/dcdsb.2014.19.2065. [27] M. Fabrizio, G. Gentili and D. W. Reynolds, On rigid heat conductors with memory, Int. J. Eng. Sci., 36 (1998), 765-782.  doi: 10.1016/S0020-7225(97)00123-7. [28] M. Fabrizio, Fractional rheological models for thermomechanical systems. Dissipation and free energies, Fract. Calc. Appl. Anal., 17 (2014), 206-223.  doi: 10.2478/s13540-014-0163-7. [29] C. Giorgi and G. Gentili, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., 51 (1993), 343-362.  doi: 10.1090/qam/1218373. [30] M. Grasselli and A. Lorenzi, Abstract nonlinear Volterra integro-differential equations with nonsmooth kernels, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9) Mat. Appl., 2 (1991), 43-53. [31] M. E. Gurtin, Modern continuum thermodynamics, Mechanics Today, 1 (1974), 168-213.  doi: 10.1016/B978-0-08-017246-0.50009-5. [32] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373. [33] J. Janno and L. von Wolfersdorf, Identification of weakly singular memory kernels in viscoelasticity, ZAMM Z. Angew. Math. Mech., 78 (1998), 391-403.  doi: 10.1002/(SICI)1521-4001(199806)78:6<391::AID-ZAMM391>3.0.CO;2-J. [34] J. Janno and L. von Wolfersdorf, Identification of weakly singular memory kernels in heat conduction, Z. Angew. Math. Mech., 77 (1997), 243-257.  doi: 10.1002/zamm.19970770403. [35] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College, Press, London, 2010.  doi: 10.1142/9781848163300. [36] M. McCarthy, Constitutive equations for thermomechanical materials with memory, Int. J. Eng. Sci., 8 (1970), 467-474.  doi: 10.1016/0020-7225(70)90023-6. [37] R. K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal., 2 (1971), 242-258.  doi: 10.1137/0502022.

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##### References:
 [1] M. J. Ablowitz and A.S. Fokas, Complex Variables, Cambridge University Press, Cambridge, 1997. [2] G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. of Mech. and Appl. Math., 57 (2004), 429-446.  doi: 10.1093/qjmam/57.3.429. [3] L. Boltzmann, Zur theorie der elastichen nachwirkung, Annalen der physik und chemie, 77 (1876), 624-654. [4] S. Carillo, Regular and singular kernel problems in magneto-viscoelasticity, Meccanica S.I. "New trends in Dynamics and Stability", 52 (2017), 3053-3060.  doi: 10.1007/s11012-017-0722-1. [5] S. Carillo, Existence, uniqueness and exponential decay: An evolution problem in heat conduction with memory, Quart. Appl. Math., 69 (2011), 635-649.  doi: 10.1090/S0033-569X-2011-01223-1. [6] S. Carillo, An evolution problem in materials with fading memory: Solution's existence and uniqueness, Complex Var. Elliptic Equ., 56 (2011), 481-492.  doi: 10.1080/17476931003786667. [7] S. Carillo, Materials with memory: Free energies & solution exponential decay, Commun. Pure Appl. Anal., 9 (2010), 1235-1248.  doi: 10.3934/cpaa.2010.9.1235. [8] S. Carillo, A 3-Dimensional Singular Kernel Problem in Viscoelasticity: An Existence Result, Atti Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali, ISSN 1825-1242 96, No. S2, A1 (2018), arXiv:1808.02411. doi: 10.1478/AAPP.96S2A1. [9] S. Carillo and C. Giorgi, Non-classical Memory Kernels in Linear Viscoelasticity, Chapter 13 in "Viscoelastic and Viscoplastic Materials", M.F. El-Amin Editor, ISBN 978-953-51-2603-4, Print ISBN 978-953-51-2602-7, Published: September 6, 2016, InTech. doi: 10.5772/64251. [10] S. Carillo, Singular kernel problems in materials with memory, Meccanica, 50 (2015), 603-615.  doi: 10.1007/s11012-014-0083-y. [11] S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, A magneto-viscoelasticity problem with a singular memory kernel, Nonlinear Analysis Series B: Real World Applications, 35 (2017), 200-210.  doi: 10.1016/j.nonrwa.2016.10.014. [12] S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, On weak regularity requirements of the relaxation modulus in viscoelasticity, arXiv: 1811.06723, (2018). [13] S. Carillo, V. Valente and G. Vergara Caffarelli, A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity, Applicable Analisys: An International Journal, 1563-504X, 90 (2011), 1791-1802. doi: 10.1080/00036811003735832. [14] S. Carillo, V. Valente and G. Vergara Caffarelli, An existence theorem for the magnetic-viscoelastic problem, Discrete and Continuous Dynamical Systems Series S., 5 (2012), 435-447.  doi: 10.3934/dcdss.2012.5.435. [15] S. Carillo, V. Valente and G. Vergara Caffarelli, Heat conduction with memory: A singular kernel problem, Evolution Equations and Control Theory, 3 (2014), 399-410.  doi: 10.3934/eect.2014.3.399. [16] S. Carillo, V. Valente and G. Vergara Caffarelli, A linear viscoelasticity problem with a singular memory kernel: An existence and uniqueness result, Differential and Integral Equations, 26 (2013), 1115-1125. [17] S. Carillo, Some remarks on materials with memory: Heat conduction and viscoelasticity, Journal of Nonlinear Mathematical Physics, 12 (2005), 163-178.  doi: 10.2991/jnmp.2005.12.s1.14. [18] S. Carillo, Some remarks on materials with memory: Heat conduction and viscoelasticity, Journal of Nonlinear Mathematical Physics, 3 (2015), i-iii.  doi: 10.1080/14029251.2014.971573. [19] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Universitá Modena, 3 (1949), 83-101. [20] J. Ciambella, A. Paolone and S. Vidoli, Memory decay rates of viscoelastic solids: Not too slow, but not too fast either, Rheol. Acta, 50 (2011), 661-674.  doi: 10.1007/s00397-011-0549-y. [21] B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal., 17 (1964), 1-46.  doi: 10.1007/BF00283864. [22] B. D. Coleman and E. H. Dill, On thermodynamics and stability of materials with memory, Arch. Rat. Mech. Anal., 51 (1973), 1-53.  doi: 10.1007/BF00275991. [23] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Diff. Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4. [24] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [25] B. de Andrade and A. Viana, On a fractional reaction diffusion equation, Z. Angew. Math. Phys., 68 (2017), Art. 59, 11 pp. doi: 10.1007/s00033-017-0801-0. [26] L. Deseri, M. Zingales and P. Pollaci, The state of fractional hereditary materials (FHM), Discrete and Continuous Dynamical Systems - B, 19 (2014), 2065-2089.  doi: 10.3934/dcdsb.2014.19.2065. [27] M. Fabrizio, G. Gentili and D. W. Reynolds, On rigid heat conductors with memory, Int. J. Eng. Sci., 36 (1998), 765-782.  doi: 10.1016/S0020-7225(97)00123-7. [28] M. Fabrizio, Fractional rheological models for thermomechanical systems. Dissipation and free energies, Fract. Calc. Appl. Anal., 17 (2014), 206-223.  doi: 10.2478/s13540-014-0163-7. [29] C. Giorgi and G. Gentili, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., 51 (1993), 343-362.  doi: 10.1090/qam/1218373. [30] M. Grasselli and A. Lorenzi, Abstract nonlinear Volterra integro-differential equations with nonsmooth kernels, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9) Mat. Appl., 2 (1991), 43-53. [31] M. E. Gurtin, Modern continuum thermodynamics, Mechanics Today, 1 (1974), 168-213.  doi: 10.1016/B978-0-08-017246-0.50009-5. [32] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373. [33] J. Janno and L. von Wolfersdorf, Identification of weakly singular memory kernels in viscoelasticity, ZAMM Z. Angew. Math. Mech., 78 (1998), 391-403.  doi: 10.1002/(SICI)1521-4001(199806)78:6<391::AID-ZAMM391>3.0.CO;2-J. [34] J. Janno and L. von Wolfersdorf, Identification of weakly singular memory kernels in heat conduction, Z. Angew. Math. Mech., 77 (1997), 243-257.  doi: 10.1002/zamm.19970770403. [35] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College, Press, London, 2010.  doi: 10.1142/9781848163300. [36] M. McCarthy, Constitutive equations for thermomechanical materials with memory, Int. J. Eng. Sci., 8 (1970), 467-474.  doi: 10.1016/0020-7225(70)90023-6. [37] R. K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal., 2 (1971), 242-258.  doi: 10.1137/0502022.
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