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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), 269-291.  Google Scholar

[2]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.  Google Scholar

[3]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  Google Scholar

[4]

B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106. doi: 10.1137/S0036141001388592.  Google Scholar

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M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. 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), B471-B473.  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), 121-133.  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), 113-126. 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, New York, 1965.  Google Scholar

[10]

T. Hillen and K. P. Hadeler, Hyperbolic systems and transport equations in mathematical biology, in "Analysis and Numerics for Conservation Laws," Springer, Berlin, (2005), 257-279. 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, Chapman & Hall/CRC, Boca Raton, FL, 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), 5231-5243. 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), 6557-6563. doi: 10.1103/PhysRevE.56.6557.  Google Scholar

[14]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.  Google Scholar

[15]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188. 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), 721-730. 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, Springer-Verlag, New York, 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), 847-857. 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), 269-291.  Google Scholar

[2]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.  Google Scholar

[3]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  Google Scholar

[4]

B. R. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106. 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), 139-152. 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), B471-B473.  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), 121-133.  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), 113-126. 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, New York, 1965.  Google Scholar

[10]

T. Hillen and K. P. Hadeler, Hyperbolic systems and transport equations in mathematical biology, in "Analysis and Numerics for Conservation Laws," Springer, Berlin, (2005), 257-279. 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, Chapman & Hall/CRC, Boca Raton, FL, 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), 5231-5243. 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), 6557-6563. doi: 10.1103/PhysRevE.56.6557.  Google Scholar

[14]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.  Google Scholar

[15]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188. 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), 721-730. 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, Springer-Verlag, New York, 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), 847-857. doi: 10.2307/1999112.  Google Scholar

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