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Exponential stability for a class of linear hyperbolic equations with hereditary memory
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 |
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. |
[2] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[3] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. |
[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. |
[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. |
[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. |
[18] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
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. |
[2] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[3] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. |
[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. |
[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. |
[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. |
[18] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
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