# American Institute of Mathematical Sciences

September  2014, 3(3): 399-410. doi: 10.3934/eect.2014.3.399

## Heat conduction with memory: A singular kernel problem

 1 Dipartimento di Scienze di Base e Applicate, per l'Ingegneria - Sezione Matematica, Sapienza Università di Roma, Rome, Italy 2 Istituto per le Applicazioni del Calcolo M. Picone, C.N.R. Consiglio Nazionale delle Ricerche, Rome, Italy

Received  May 2013 Revised  February 2014 Published  August 2014

The existence and uniqueness of solution to an integro-differential problem arising in heat conduction with memory is here considered. Specifically, a singular kernel problem is analyzed in the case of a multi-dimensional rigid heat conductor. The choice to investigate a singular kernel material is suggested by applications to model a wider variety of materials and, in particular, new materials whose heat flux relaxation function may be superiorly unbounded at the initial time $t=0$. The present study represents a generalization to higher dimensions of a previous one concerning a $1$-dimensional problem in the framework of linear viscoelasticity with memory. Specifically, an existence theorem is here proved when initial homogeneous data are assumed. Indeed, the choice of homogeneous data is needed to obtain the a priori estimate in Section 2 on which the subsequent results, are based.
Citation: Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. Heat conduction with memory: A singular kernel problem. Evolution Equations and Control Theory, 2014, 3 (3) : 399-410. doi: 10.3934/eect.2014.3.399
##### References:
 [1] 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. [2] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0. [3] 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, First published on 19 August 2010, 90 (2011), 1791-1802. doi: 10.1080/00036811003735832. [4] 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. [5] 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. doi: http://projecteuclid.org/euclid.die/1372858565. [6] S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, in preparation, (2014). [7] S. Carillo, Some remarks on materials with memory: heat conduction and viscoelasticity, Journal of Nonlinear Mathematical Physics Supplement 1, 12 (2005), 163-178. doi: 10.2991/jnmp.2005.12.s1.14. [8] S. Carillo, Evolution problems in materials with fading memory, Matematiche (Catania), 62 (2007), 93-105. doi: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/30/29. [9] S. Carillo, An evolution problem in materials with fading memory: Solution's existence and uniqueness, Complex Variables and Elliptic Equations An International Journal, 56 (2011), 481-492. doi: 10.1080/17476931003786667. [10] S. Carillo, Materials with mMemory: Free energies & solutions' exponential decay, Commun. Pure Appl. Anal., 9 (2010), 1235-1248. doi: 10.3934/cpaa.2010.9.1235. [11] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Universitá Modena, 3 (1948), 83-101. [12] V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptotic Analysis, 50 (2006), 269-291. [13] M. Chipot, I. Shafrir, V. Valente and G. Vergara Caffarelli, A nonlocal problem arising in the study of magneto-elastic interactions, Boll. UMI Serie IX, I (2008), 197-222. [14] M. Chipot, I. Shafrir, V. Valente and G. Vergara Caffarelli, On a hyperbolic-parabolic system arising in magneto-elasticity, J. Math. Anal. Appl., 352 (2009), 120-131. doi: 10.1016/j.jmaa.2008.04.013. [15] B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal., 17 (1964), 1-46. doi: 10.1007/BF00283864. [16] 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. [17] 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. [18] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. [19] 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. [20] M. Fabrizio, B. Lazzari and A. Morro, Mathematical Models and Methods for Smart Materials, Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co., Inc., River Edge, NJ, 62, 2002. doi: 10.1142/5162. [21] C. Giorgi and G. Gentili, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., 51 (1993), 343-62. [22] C. Giorgi and V. Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, Nonlinear Differential Equations and Applications, 8 (2001), 157-171. doi: 10.1007/PL00001443. [23] 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. [24] M. E. Gurtin, Modern Continuum Thermodynamics, Mechanics Today, 1 (1972), 168-213. [25] 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. [26] 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>3.3.CO;2-A. [27] 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. [28] 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. [29] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College Press, London, 2010. doi: 10.1142/9781848163300. [30] E. Mainini and G. Mola, Exponential and polynomial decay for first order linear Volterra evolution equations, Quart. Appl. Math., 67 (2009), 93-111. [31] B. Miara, G. Stavroulakis and V. Valente, Topics on Mathematics for Smart Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. [32] 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. [33] N.-E. Tatar, Exponential decay for a viscoelastic problem with a singular kernel, Zeitschrift fur Angewandte Mathematik und Physik, 60 (2009), 640-650. doi: 10.1007/s00033-008-8030-1. [34] V. Valente and G. Vergara Caffarelli, On the dynamics of magneto-elastic interactions: Existence of solutions and limit behavior, Asymptotic Analysis, 51 (2007), 319-333. [35] G. Vergara Caffarelli, Dissipativity and uniqueness for the one-dimensional dynamical problem of linear viscoelasticity, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82 (1990), 483-488. [36] G. Vergara Caffarelli, Dissipativity and existence for the one-dimensional dynamical problem of linear viscoelasticity, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82 (1988), 489-496. [37] S. T. Wu, Exponential decay for a nonlinear viscoelastic equation with singular kernels, Acta. Mathematica Scientia, 32 (2012), 2237-2246. doi: 10.1016/S0252-9602(12)60173-8.

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##### References:
 [1] 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. [2] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0. [3] 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, First published on 19 August 2010, 90 (2011), 1791-1802. doi: 10.1080/00036811003735832. [4] 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. [5] 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. doi: http://projecteuclid.org/euclid.die/1372858565. [6] S. Carillo, M. Chipot, V. Valente and G. Vergara Caffarelli, in preparation, (2014). [7] S. Carillo, Some remarks on materials with memory: heat conduction and viscoelasticity, Journal of Nonlinear Mathematical Physics Supplement 1, 12 (2005), 163-178. doi: 10.2991/jnmp.2005.12.s1.14. [8] S. Carillo, Evolution problems in materials with fading memory, Matematiche (Catania), 62 (2007), 93-105. doi: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/30/29. [9] S. Carillo, An evolution problem in materials with fading memory: Solution's existence and uniqueness, Complex Variables and Elliptic Equations An International Journal, 56 (2011), 481-492. doi: 10.1080/17476931003786667. [10] S. Carillo, Materials with mMemory: Free energies & solutions' exponential decay, Commun. Pure Appl. Anal., 9 (2010), 1235-1248. doi: 10.3934/cpaa.2010.9.1235. [11] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Universitá Modena, 3 (1948), 83-101. [12] V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptotic Analysis, 50 (2006), 269-291. [13] M. Chipot, I. Shafrir, V. Valente and G. Vergara Caffarelli, A nonlocal problem arising in the study of magneto-elastic interactions, Boll. UMI Serie IX, I (2008), 197-222. [14] M. Chipot, I. Shafrir, V. Valente and G. Vergara Caffarelli, On a hyperbolic-parabolic system arising in magneto-elasticity, J. Math. Anal. Appl., 352 (2009), 120-131. doi: 10.1016/j.jmaa.2008.04.013. [15] B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal., 17 (1964), 1-46. doi: 10.1007/BF00283864. [16] 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. [17] 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. [18] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. [19] 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. [20] M. Fabrizio, B. Lazzari and A. Morro, Mathematical Models and Methods for Smart Materials, Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co., Inc., River Edge, NJ, 62, 2002. doi: 10.1142/5162. [21] C. Giorgi and G. Gentili, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., 51 (1993), 343-62. [22] C. Giorgi and V. Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, Nonlinear Differential Equations and Applications, 8 (2001), 157-171. doi: 10.1007/PL00001443. [23] 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. [24] M. E. Gurtin, Modern Continuum Thermodynamics, Mechanics Today, 1 (1972), 168-213. [25] 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. [26] 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>3.3.CO;2-A. [27] 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. [28] 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. [29] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College Press, London, 2010. doi: 10.1142/9781848163300. [30] E. Mainini and G. Mola, Exponential and polynomial decay for first order linear Volterra evolution equations, Quart. Appl. Math., 67 (2009), 93-111. [31] B. Miara, G. Stavroulakis and V. Valente, Topics on Mathematics for Smart Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. [32] 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. [33] N.-E. Tatar, Exponential decay for a viscoelastic problem with a singular kernel, Zeitschrift fur Angewandte Mathematik und Physik, 60 (2009), 640-650. doi: 10.1007/s00033-008-8030-1. [34] V. Valente and G. Vergara Caffarelli, On the dynamics of magneto-elastic interactions: Existence of solutions and limit behavior, Asymptotic Analysis, 51 (2007), 319-333. [35] G. Vergara Caffarelli, Dissipativity and uniqueness for the one-dimensional dynamical problem of linear viscoelasticity, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82 (1990), 483-488. [36] G. Vergara Caffarelli, Dissipativity and existence for the one-dimensional dynamical problem of linear viscoelasticity, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82 (1988), 489-496. [37] S. T. Wu, Exponential decay for a nonlinear viscoelastic equation with singular kernels, Acta. Mathematica Scientia, 32 (2012), 2237-2246. doi: 10.1016/S0252-9602(12)60173-8.
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