Existence and uniqueness for an integro-differential equation with singular kernel

Valeria Berti

Bollettino dell'Unione Matematica Italiana (2006)

  • Volume: 9-B, Issue: 2, page 299-309
  • ISSN: 0392-4033

Abstract

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In this paper we study the evolutive problem of linear viscoelasticity with a singular kernel memory G . We assume that G presents an initial singularity, so that it is not a L 1 -function in time, whereas the relaxation function G is integrable at t = 0 . By applying the Fourier transform method, we prove a theorem of existence and uniqueness of the weak solutions in a functional space whose definition is strictly related to the properties of the kernel memory.

How to cite

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Berti, Valeria. "Existence and uniqueness for an integro-differential equation with singular kernel." Bollettino dell'Unione Matematica Italiana 9-B.2 (2006): 299-309. <http://eudml.org/doc/289611>.

@article{Berti2006,
abstract = {In this paper we study the evolutive problem of linear viscoelasticity with a singular kernel memory $G'$. We assume that $G'$ presents an initial singularity, so that it is not a $L^1$-function in time, whereas the relaxation function $G$ is integrable at $t = 0$. By applying the Fourier transform method, we prove a theorem of existence and uniqueness of the weak solutions in a functional space whose definition is strictly related to the properties of the kernel memory.},
author = {Berti, Valeria},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {299-309},
publisher = {Unione Matematica Italiana},
title = {Existence and uniqueness for an integro-differential equation with singular kernel},
url = {http://eudml.org/doc/289611},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Berti, Valeria
TI - Existence and uniqueness for an integro-differential equation with singular kernel
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/6//
PB - Unione Matematica Italiana
VL - 9-B
IS - 2
SP - 299
EP - 309
AB - In this paper we study the evolutive problem of linear viscoelasticity with a singular kernel memory $G'$. We assume that $G'$ presents an initial singularity, so that it is not a $L^1$-function in time, whereas the relaxation function $G$ is integrable at $t = 0$. By applying the Fourier transform method, we prove a theorem of existence and uniqueness of the weak solutions in a functional space whose definition is strictly related to the properties of the kernel memory.
LA - eng
UR - http://eudml.org/doc/289611
ER -

References

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  1. FABRIZIO, M. - LAZZARI, B., On the existence and asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rational Mech. Anal., 116 (2) (1991), 139-152. Zbl0766.73013
  2. FABRIZIO, M. - LAZZARI, B., The domain of dependence inequality and asymptotic stability for a viscoelastic solid, Nonlinear Oscil., 1 (1998), 117-133. Zbl1071.74587
  3. FABRIZIO, M. - MORRO, A., Mathematical problems in linear viscoelasticity, SIAM, Philadelphia, 1992. Zbl0753.73003
  4. GENTILI, G., Regularity and stability for a viscoelastic material with a singular memory kernel, J. Elasticity, 37 (2) (1995), 139-156. Zbl0818.73026
  5. HANYGA, A., Wave propagation in media with singular memory, Math. Comput. Modelling, 34 (12-13) (2001), 1329-1421. Zbl1011.74033
  6. HRUSA, W.J. - RENARDY, M., On wave propagation in linear viscoelasticity, Quart. Appl. Math., 43 (2) (1985), 237-254. Zbl0571.73026
  7. LADYZHENSKAYA, O. A., The boundary value problem of mathematical physics, Springer, New York, 1985. Zbl0588.35003
  8. RENARDY, M. - HRUSA, W. J. - NOHEL, J. A., Mathematical problems in viscoelasticity, Longman Scientific & Technical, John Wiley & Sons, New York, 1987. 
  9. SHOWALTER, R. E., Hilbert space methods for differential equations, Pitman, London, 1977. Zbl0364.35001
  10. TREVES, F., Basic linear partial differential equations, Acad. press, New York, 1975. Zbl0305.35001

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