Stability analysis for linear heat conduction with memory kernels described by Gamma functions

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August 2015, 35(8): 3569-3584. doi: 10.3934/dcds.2015.35.3569

Stability analysis for linear heat conduction with memory kernels described by Gamma functions

Corrado Mascia ^1,

1.	Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale A. Moro, 2, I–00185 Roma

Received October 2014 Revised November 2014 Published February 2015

Abstract
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This paper analyzes heat equation with memory in the case of kernels that are linear combinations of Gamma distributions. In this case, it is possible to rewrite the non-local equation as a local system of partial differential equations of hyperbolic type. Stability is studied in details by analyzing the corresponding dispersion relation, providing sufficient stability condition for the general case and sharp instability thresholds in the case of linear combinations of the first three Gamma functions.

Keywords: memory kernels, Diffusion equation, stability., heat conduction.

Mathematics Subject Classification: Primary: 74D05; Secondary: 35L45, 35B35, 76R5.

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

References:

[1]	C. Cattaneo, Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena, 3 (1949), 83-101.
[2]	V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291.
[3]	M. Conti, E. M. Marchini and V. Pata, Exponential stability for a class of linear hyperbolic equations with hereditary memory, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1555-1565. doi: 10.3934/dcdsb.2013.18.1555.
[4]	A. Chowdury and C. I. Christov, Memory effects for the heat conductivity of random suspensions of spheres, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 3253-3273. doi: 10.1098/rspa.2010.0133.
[5]	B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106 (electronic). doi: 10.1137/S0036141001388592.
[6]	G. Fichera, Avere una memoria tenace crea gravi problemi, (Italian) Arch. Rational Mech. Anal., 70 (1979), 101-112. doi: 10.1007/BF00250347.
[7]	C. Giorgi, M. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: A semigroup approach, Commun. Appl. Anal., 5 (2001), 121-133.
[8]	M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.
[9]	D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41.
[10]	W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188. doi: 10.1137/0146013.
[11]	Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.
[12]	H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

show all references

References:

[1]	C. Cattaneo, Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena, 3 (1949), 83-101.
[2]	V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269-291.
[3]	M. Conti, E. M. Marchini and V. Pata, Exponential stability for a class of linear hyperbolic equations with hereditary memory, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1555-1565. doi: 10.3934/dcdsb.2013.18.1555.
[4]	A. Chowdury and C. I. Christov, Memory effects for the heat conductivity of random suspensions of spheres, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 3253-3273. doi: 10.1098/rspa.2010.0133.
[5]	B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106 (electronic). doi: 10.1137/S0036141001388592.
[6]	G. Fichera, Avere una memoria tenace crea gravi problemi, (Italian) Arch. Rational Mech. Anal., 70 (1979), 101-112. doi: 10.1007/BF00250347.
[7]	C. Giorgi, M. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: A semigroup approach, Commun. Appl. Anal., 5 (2001), 121-133.
[8]	M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.
[9]	D. D. Joseph and L. Preziosi, Heat waves, Rev. Modern Phys., 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41.
[10]	W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188. doi: 10.1137/0146013.
[11]	Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.
[12]	H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

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