Decay rates of Volterra equations on ℝN

Monica Conti; Stefania Gatti; Vittorino Pata

Open Mathematics (2007)

  • Volume: 5, Issue: 4, page 720-732
  • ISSN: 2391-5455

Abstract

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This note is concerned with the linear Volterra equation of hyperbolic type ∂ t t u ( t ) - α Δ u ( t ) + ∫ 0 t μ ( s ) Δ u ( t - s ) d s = 0 on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.

How to cite

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Monica Conti, Stefania Gatti, and Vittorino Pata. "Decay rates of Volterra equations on ℝN." Open Mathematics 5.4 (2007): 720-732. <http://eudml.org/doc/269324>.

@article{MonicaConti2007,
abstract = {This note is concerned with the linear Volterra equation of hyperbolic type \[\partial \_\{tt\} u(t) - \alpha \Delta u(t) + \int \_0^t \{\mu (s)\Delta u(t - s)\} ds = 0\] on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.},
author = {Monica Conti, Stefania Gatti, Vittorino Pata},
journal = {Open Mathematics},
keywords = {Integro-differential equations; memory kernel; polynomial decay; decay of energy},
language = {eng},
number = {4},
pages = {720-732},
title = {Decay rates of Volterra equations on ℝN},
url = {http://eudml.org/doc/269324},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Monica Conti
AU - Stefania Gatti
AU - Vittorino Pata
TI - Decay rates of Volterra equations on ℝN
JO - Open Mathematics
PY - 2007
VL - 5
IS - 4
SP - 720
EP - 732
AB - This note is concerned with the linear Volterra equation of hyperbolic type \[\partial _{tt} u(t) - \alpha \Delta u(t) + \int _0^t {\mu (s)\Delta u(t - s)} ds = 0\] on the whole space ℝN. New results concerning the decay of the associated energy as time goes to infinity were established.
LA - eng
KW - Integro-differential equations; memory kernel; polynomial decay; decay of energy
UR - http://eudml.org/doc/269324
ER -

References

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  1. [1] M. Conti, S. Gatti, V. Pata: “Uniform decay properties of linear Volterra integrodifferential equations”, Math. Models Methods Appl. Sci. (to appear). Zbl1152.35318
  2. [2] C.M. Dafermos: “An abstract Volterra equation with applications to linear viscoelasticity”, J. Differential Equations, Vol. 7, (1970), pp. 554–569. http://dx.doi.org/10.1016/0022-0396(70)90101-4 
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  9. [9] Z. Liu, S. Zheng: “On the exponential stability of linear viscoelasticity and thermoviscoelasticity”, Quart. Appl. Math., Vol. 54, (1996), pp. 21–31. Zbl0868.35011
  10. [10] J.E. Muñoz Rivera: “Asymptotic behaviour in linear viscoelasticity”, Quart. Appl. Math., Vol. 52, (1994), pp. 629–648. Zbl0814.35009
  11. [11] J.E. Muñoz Rivera, E. Cabanillas Lapa: “Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels”, Comm. Math. Phys., Vol. 177, (1996), pp. 583–602. http://dx.doi.org/10.1007/BF02099539 Zbl0852.73026
  12. [12] V. Pata: “Exponential stability in linear viscoelasticity”, Quart. Appl. Math., Vol. 64, (2006), pp. 499–513. Zbl1117.35052
  13. [13] V. Pata, A. Zucchi: “Attractors for a damped hyperbolic equation with linear memory”, Adv. Math. Sci. Appl., Vol. 11, (2001), pp. 505–529. Zbl0999.35014
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