Abstract
In this survey paper, we discuss the decay properties of the semigroup generated by a linear integro-differential equation in a Hilbert space, which is an abstract version of the equation
describing the dynamics of linearly viscoelastic bodies.
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Chepyzhov V.V., Mainini E., Pata V.: Stability of abstract linear semigroups arising from heat conduction with memory. Asymptot. Anal. 46, 251–273 (2006)
Chepyzhov V.V., Pata V.: Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptot. Anal. 50, 269–291 (2006)
Conti M., Gatti S., Pata V.: Uniform decay properties of linear Volterra integro-differential equations. Math. Models Methods Appl. Sci. 18, 1–21 (2008)
Curtain R.F., Zwart H.J.: An introduction to infinite-dimensional linear system theory. Springer, New York (1995)
Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)
C.M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, in ��Applications of Methods of Functional Analysis to Problems in Mechanics” (P. Germain and B. Nayroles, Eds.), pp.295–306, Lecture Notes in Mathematics no.503, Springer-Verlag, Berlin-New York, 1976.
Fabrizio M., Lazzari B.: On the existence and asymptotic stability of solutions for linear viscoelastic solids. Arch. Rational Mech. Anal. 116, 139–152 (1991)
M. Fabrizio, A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics no.12, SIAM, Philadelphia, 1992.
Fabrizio M., Polidoro S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81, 1245–1264 (2002)
C. Giorgi, B. Lazzari, Uniqueness and stability in linear viscoelasticity: some counterexamples, in “Waves and stability in continuous media (Sorrento, 1989)” pp.146–153, Ser. Adv. Math. Appl. Sci. no.4, World Sci. Publishing, River Edge, NJ, 1991.
Giorgi C., Lazzari B.: On the stability for linear viscoelastic solids. Quart. Appl. Math. 55, 659–675 (1997)
Giorgi C., Muñoz Rivera J.E., Pata V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)
Grasselli M., Muñoz Rivera J.E., Pata V.: On the decay of the linear thermoelastic plate with memory. J. Math. Anal. Appl. 309, 1–14 (2005)
M. Grasselli, V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp.155–178, Progr. Nonlinear Differential Equations Appl. no.50, Birkhäuser, Boston, 2002.
Liu Z., Zheng S.: On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. 54, 21–31 (1996)
Z. Liu, S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics no.398, Chapman & Hall/CRC, Boca Raton, FL, 1999.
Muñoz Rivera J.E.: Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. 52, 629–648 (1994)
Pata V.: Exponential stability in linear viscoelasticity. Quart. Appl. Math. 64, 499–513 (2006)
Pata V., Zucchi A.: Attractors for a damped hyperbolic equation with linear memory. Adv. Math. Sci. Appl. 11, 505–529 (2001)
Pazy A.: Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983)
Prüss J.: On the spectrum of C 0-semigroups. Trans. Amer. Math. Soc. 284, 847–857 (1984)
Renardy M., Hrusa W.J., Nohel J.A.: Mathematical problems in viscoelasticity. Harlow John Wiley & Sons, Inc., New York (1987)
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Dedicated to the memory of Professor Giovanni Prouse
Work partially supported by the Italian PRIN Research Project 2006 Problemi a frontiera libera, transizioni di fase e modelli di isteresi.
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Pata, V. Stability and exponential stability in linear viscoelasticity. Milan J. Math. 77, 333–360 (2009). https://doi.org/10.1007/s00032-009-0098-3
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DOI: https://doi.org/10.1007/s00032-009-0098-3