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

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

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.
Citation: Corrado Mascia. Stability analysis for linear heat conduction with memory kernels described by Gamma functions. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3569-3584. doi: 10.3934/dcds.2015.35.3569
References:
[1]

Atti Semin. Mat. Fis. Univ. Modena, 3 (1949), 83-101.  Google Scholar

[2]

Asymptot. Anal., 50 (2006), 269-291.  Google Scholar

[3]

Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1555-1565. doi: 10.3934/dcdsb.2013.18.1555.  Google Scholar

[4]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 3253-3273. doi: 10.1098/rspa.2010.0133.  Google Scholar

[5]

SIAM J. Math. Anal., 33 (2002), 1090-1106 (electronic). doi: 10.1137/S0036141001388592.  Google Scholar

[6]

Arch. Rational Mech. Anal., 70 (1979), 101-112. doi: 10.1007/BF00250347.  Google Scholar

[7]

Commun. Appl. Anal., 5 (2001), 121-133.  Google Scholar

[8]

Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.  Google Scholar

[9]

Rev. Modern Phys., 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41.  Google Scholar

[10]

SIAM J. Appl. Math., 46 (1986), 171-188. doi: 10.1137/0146013.  Google Scholar

[11]

Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.  Google Scholar

[12]

Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

show all references

References:
[1]

Atti Semin. Mat. Fis. Univ. Modena, 3 (1949), 83-101.  Google Scholar

[2]

Asymptot. Anal., 50 (2006), 269-291.  Google Scholar

[3]

Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1555-1565. doi: 10.3934/dcdsb.2013.18.1555.  Google Scholar

[4]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 3253-3273. doi: 10.1098/rspa.2010.0133.  Google Scholar

[5]

SIAM J. Math. Anal., 33 (2002), 1090-1106 (electronic). doi: 10.1137/S0036141001388592.  Google Scholar

[6]

Arch. Rational Mech. Anal., 70 (1979), 101-112. doi: 10.1007/BF00250347.  Google Scholar

[7]

Commun. Appl. Anal., 5 (2001), 121-133.  Google Scholar

[8]

Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.  Google Scholar

[9]

Rev. Modern Phys., 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41.  Google Scholar

[10]

SIAM J. Appl. Math., 46 (1986), 171-188. doi: 10.1137/0146013.  Google Scholar

[11]

Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.  Google Scholar

[12]

Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

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