American Institute of Mathematical Sciences

Advanced
August  2013, 18(6): 1555-1565. doi: 10.3934/dcdsb.2013.18.1555

Exponential stability for a class of linear hyperbolic equations with hereditary memory

Monica Conti 1, , Elsa M. Marchini 2, and  Vittorino Pata 1,

1. 

Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano
2. 

Politecnico di Milano - Dipartimento di Matematica “F. Brioschi”, Via Bonardi 9, 20133 Milano, Italy

Received  June 2011 Revised  November 2011 Published  March 2013

We establish a necessary and sufficient condition of exponential stability for the contraction semigroup generated by an abstract version of the linear differential equation $$∂_t u(t)-\int_0^\infty k(s)\Delta u(t-s)ds = 0 $$ modeling hereditary heat conduction of Gurtin-Pipkin type.
Keywords: semigroups of linear contractions, exponential stability., memory kernels, Hereditary heat conduction.
Mathematics Subject Classification: 35B40, 45K05, 45M10, 47D0.
Citation: Monica Conti, Elsa M. Marchini, Vittorino Pata. Exponential stability for a class of linear hyperbolic equations with hereditary memory. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1555-1565. doi: 10.3934/dcdsb.2013.18.1555
References:
[1]

V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory,, Asymptot. Anal., 50 (2006), 269.   Google Scholar

[2]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity,, Asymptot. Anal., 46 (2006), 251.   Google Scholar

[3]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297.   Google Scholar

[4]

B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process,, SIAM J. Math. Anal., 33 (2002), 1090.  doi: 10.1137/S0036141001388592.  Google Scholar

[5]

M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids,, Arch. Rational Mech. Anal., 116 (1991), 139.  doi: 10.1007/BF00375589.  Google Scholar

[6]

D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles),, C. R. Acad. Sci. Paris Sér. A-B, 277 (1973).   Google Scholar

[7]

C. Giorgi, M. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: a semigroup approach,, Comm. Appl. Anal., 5 (2001), 121.   Google Scholar

[8]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed,, Arch. Rational Mech. Anal., 31 (1968), 113.  doi: 10.1007/BF00281373.  Google Scholar

[9]

E. Hewitt and K. Stromberg, "Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable,", Springer-Verlag, (1965).   Google Scholar

[10]

T. Hillen and K. P. Hadeler, Hyperbolic systems and transport equations in mathematical biology,, in, (2005), 257.  doi: 10.1007/3-540-27907-5_11.  Google Scholar

[11]

Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems,", Chapman & Hall/CRC Research Notes in Mathematics, 398 (1999).   Google Scholar

[12]

V. Méndez, J. Fort and J. Farjas, Speed of wave-front solutions to hyperbolic reaction-diffusion equations,, Phys. Rev. E (3), 60 (1999), 5231.  doi: 10.1103/PhysRevE.60.5231.  Google Scholar

[13]

V. Méndez and J. E. Llebot, Hyperbolic reaction-diffusion equations for a forest fire model,, Phys. Rev. E (3), 56 (1997), 6557.  doi: 10.1103/PhysRevE.56.6557.  Google Scholar

[14]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity,, Quart. Appl. Math., 52 (1994), 628.   Google Scholar

[15]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory,, SIAM J. Appl. Math., 46 (1986), 171.  doi: 10.1137/0146013.  Google Scholar

[16]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels,, Commun. Pure Appl. Anal., 9 (2010), 721.  doi: 10.3934/cpaa.2010.9.721.  Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[18]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

show all references

References:
[1]

V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory,, Asymptot. Anal., 50 (2006), 269.   Google Scholar

[2]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity,, Asymptot. Anal., 46 (2006), 251.   Google Scholar

[3]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297.   Google Scholar

[4]

B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process,, SIAM J. Math. Anal., 33 (2002), 1090.  doi: 10.1137/S0036141001388592.  Google Scholar

[5]

M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids,, Arch. Rational Mech. Anal., 116 (1991), 139.  doi: 10.1007/BF00375589.  Google Scholar

[6]

D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles),, C. R. Acad. Sci. Paris Sér. A-B, 277 (1973).   Google Scholar

[7]

C. Giorgi, M. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: a semigroup approach,, Comm. Appl. Anal., 5 (2001), 121.   Google Scholar

[8]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed,, Arch. Rational Mech. Anal., 31 (1968), 113.  doi: 10.1007/BF00281373.  Google Scholar

[9]

E. Hewitt and K. Stromberg, "Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable,", Springer-Verlag, (1965).   Google Scholar

[10]

T. Hillen and K. P. Hadeler, Hyperbolic systems and transport equations in mathematical biology,, in, (2005), 257.  doi: 10.1007/3-540-27907-5_11.  Google Scholar

[11]

Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems,", Chapman & Hall/CRC Research Notes in Mathematics, 398 (1999).   Google Scholar

[12]

V. Méndez, J. Fort and J. Farjas, Speed of wave-front solutions to hyperbolic reaction-diffusion equations,, Phys. Rev. E (3), 60 (1999), 5231.  doi: 10.1103/PhysRevE.60.5231.  Google Scholar

[13]

V. Méndez and J. E. Llebot, Hyperbolic reaction-diffusion equations for a forest fire model,, Phys. Rev. E (3), 56 (1997), 6557.  doi: 10.1103/PhysRevE.56.6557.  Google Scholar

[14]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity,, Quart. Appl. Math., 52 (1994), 628.   Google Scholar

[15]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory,, SIAM J. Appl. Math., 46 (1986), 171.  doi: 10.1137/0146013.  Google Scholar

[16]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels,, Commun. Pure Appl. Anal., 9 (2010), 721.  doi: 10.3934/cpaa.2010.9.721.  Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[18]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

[1]

Corrado Mascia. Stability analysis for linear heat conduction with memory kernels described by Gamma functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3569-3584. doi: 10.3934/dcds.2015.35.3569
[2]

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

Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. Heat conduction with memory: A singular kernel problem. Evolution Equations & Control Theory, 2014, 3 (3) : 399-410. doi: 10.3934/eect.2014.3.399
[4]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[5]

Alexander Pimenov, Dmitrii I. Rachinskii. Linear stability analysis of systems with Preisach memory. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 997-1018. doi: 10.3934/dcdsb.2009.11.997
[6]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden, Adele Manes. Energy stability for thermo-viscous fluids with a fading memory heat flux. Evolution Equations & Control Theory, 2015, 4 (3) : 265-279. doi: 10.3934/eect.2015.4.265
[7]

Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321
[8]

Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148
[9]

Martin Fraas, David Krejčiřík, Yehuda Pinchover. On some strong ratio limit theorems for heat kernels. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 495-509. doi: 10.3934/dcds.2010.28.495
[10]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061
[11]

Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165
[12]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078
[13]

Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133
[14]

Fausto Ferrari, Michele Miranda Jr, Diego Pallara, Andrea Pinamonti, Yannick Sire. Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 477-491. doi: 10.3934/dcdss.2018026
[15]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[16]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203
[17]

Yuming Qin, T. F. Ma, M. M. Cavalcanti, D. Andrade. Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid. Communications on Pure & Applied Analysis, 2005, 4 (3) : 635-664. doi: 10.3934/cpaa.2005.4.635
[18]

Monica Conti, Stefania Gatti, Alain Miranville. A singular cahn-hilliard-oono phase-field system with hereditary memory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3033-3054. doi: 10.3934/dcds.2018132
[19]

John R. Tucker. Attractors and kernels: Linking nonlinear PDE semigroups to harmonic analysis state-space decomposition. Conference Publications, 2001, 2001 (Special) : 366-370. doi: 10.3934/proc.2001.2001.366
[20]

Luc Miller. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1465-1485. doi: 10.3934/dcdsb.2010.14.1465

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences