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

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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

Abstract
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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

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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

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