Decadimento uniforme per equazioni integro-differenziali lineari di Volterra

Authors

  • Stefania Gatti Università di Modena - Reggio Emilia

DOI:

https://doi.org/10.6092/issn.2240-2829/2669

Abstract

This talk is devoted to some recent results concerning the exponential and the polynomial decays of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation describing the motion of a linearly viscoelastic solid occupying a (bounded) volume at rest.
We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel. A similar analysis is carried on in the whole N-dimensional real space, although both the polynomial and the exponential decay of the memory kernel lead to a polynomial decay of the energy, with a rate influenced by the space dimension N. These results are contained in two joint papers with Monica Conti and Vittorino Pata (Politecnico di Milano).

References

F. Ammar-Khodja, A. Benabdallah, J.E. Munoz Rivera, R. Racke. Energy decay for Timoshenko systems of memory type. J. Differential Equations (Amsterdam), 194 (2003) 82–115.

J.A.D. Appleby, M. Fabrizio, B. Lazzari, D.W. Reynolds. On exponential asymptotic stability in linear viscoelasticity. Math.Models Methods Appl. Sci. (Singapore), 16 (2006) 1677–1694.

J.A.D. Appleby, D.W. Reynolds. On necessary and sufficient conditions for exponential stability in linear Volterra integro-differential equations. J. Integral Equations Appl. (Tempe), 16 (2004)

–240.

M. Conti, S. Gatti, V. Pata. Uniform decay properties of linear Volterra integro-differential equations. Math. Mod. Meth. Appl. Sci. (Singapore), 18 (2008) 21-45.

M. Conti, S. Gatti, V. Pata. Decay rates of Volterra equations on RN. Cent. Eur. J. Math. (Warsaw), 5 (2007) 720-732.

C.M. Dafermos. An abstract Volterra equation with applications to linear viscoelasticity. J. Differential Equations (Amsterdam), 7 (1970) 554-569.

C.M. Dafermos. Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. (Heidelberg), 37 (1970) 297–308.

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).

M. Fabrizio, C. Giorgi, V. Pata. A new approach to equations with memory. Arch. Ration. Mech. Anal. (Heidelberg), 198 (2010) 189-232.

M. Fabrizio, B. Lazzari. On the existence and asymptotic stability of solutions for linear viscoelastic solids. Arch. Rational Mech. Anal. (Heidelberg), 116 (1991) 139-152.

M. Fabrizio, A. Morro. Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics no.12, SIAM, Philadelphia (1992).

M. Fabrizio, S. Polidoro. Asymptotic decay for some differential systems with fading memory. Appl. Anal. (Abingdon), 81 (2002) 1245-1264.

C. Giorgi, J.E. Mu˜noz Rivera, V. Pata. Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. (Amsterdam), 260 2001, 83-99.

Z. Liu, S. Zheng. On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. (Providence), 54(1996) 21–31.

J.E. Munoz Rivera. Asymptotic behaviour in linear viscoelasticity. Quart. Appl. Math. (Providence), 52(1994) 629–648.

J.E. Munoz Rivera, E. Cabanillas Lapa. Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels. Commun. Math. Phys.

(Berlin), 177 (1996) 583-602.

J.E. Munoz Rivera, M.G. Naso. On the decay of the energy for systems with memory and indefinite dissipation. Asymptot. Anal. (Amsterdam), 49 (2006) 189–204.

J.E. Munoz Rivera, M.G. Naso, E. Vuk. Asymptotic behaviour of the energy for electromagnetic systems with memory. Math. Methods Appl. Sci. (Chichester), 27 (2004) 819–841.

J.E. Munoz Rivera, R. Racke. Magneto-thermo-elasticity – large-time behavior for linear systems. Adv. Differential Equations (Athens), 6 (2001) 359–384.

S. Murakami. Exponential asymptotic stability for scalar linear Volterra equations. Differential Integral Equations (Athens), 4 (1991) 519–525.

V. Pata. Exponential stability in linear viscoelasticity. Quart. Appl. Math. (Providence), 64 (2006) 499-513.

V. Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Commun. Pure Appl. Anal. (Springfield), 9 (2010) 721-730.

Published

2011-12-31

How to Cite

Gatti, S. (2011). Decadimento uniforme per equazioni integro-differenziali lineari di Volterra. Bruno Pini Mathematical Analysis Seminar, 2(1). https://doi.org/10.6092/issn.2240-2829/2669

Issue

Section

Articles