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 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.

[1]

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

[2]

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

[3]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems and Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002

[4]

Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure and Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721

[5]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221

[6]

Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations and Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35

[7]

Akram Ben Aissa. Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 983-993. doi: 10.3934/dcdss.2021106

[8]

M. Carme Leseduarte, Ramon Quintanilla. Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1221-1235. doi: 10.3934/cpaa.2013.12.1221

[9]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079

[10]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden, Adele Manes. Energy stability for thermo-viscous fluids with a fading memory heat flux. Evolution Equations and Control Theory, 2015, 4 (3) : 265-279. doi: 10.3934/eect.2015.4.265

[11]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[12]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[13]

Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations and Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241

[14]

Martin Fraas, David Krejčiřík, Yehuda Pinchover. On some strong ratio limit theorems for heat kernels. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 495-509. doi: 10.3934/dcds.2010.28.495

[15]

Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control and Related Fields, 2021, 11 (4) : 965-985. doi: 10.3934/mcrf.2020054

[16]

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

[17]

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

[18]

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

[19]

Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277

[20]

Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic and Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]