Geodesic
Existence and uniqueness for an integro-differential equation with singular kernel
Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 2, pp. 299-309.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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.
In questo articolo si studia un problema evolutivo per la viscoelasticità lineare, supponendo che il nucleo di memoria $G'$ sia singolare. Si assume che $G'$ presenti una singolarità iniziale in modo che non sia una funzione $L^1$ nel tempo, ma che la funzione $G$ sia integrabile per $t = 0$. Applicando il metodo delle trasformate di Fourier, si dimostra un teorema di esistenza e unicità della soluzione debole, in un opportuno spazio funzionale, la cui definizione dipende esplicitamente dalle proprietà del nucleo di memoria. 
@article{BUMI_2006_8_9B_2_a3,
     author = {Berti, Valeria},
     title = {Existence and uniqueness for an integro-differential equation with singular kernel},
     journal = {Bollettino della Unione matematica italiana},
     pages = {299--309},
     publisher = {mathdoc},
     volume = {Ser. 8, 9B},
     number = {2},
     year = {2006},
     zbl = {1178.45011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_2_a3/}
}
TY  - JOUR
AU  - Berti, Valeria
TI  - Existence and uniqueness for an integro-differential equation with singular kernel
JO  - Bollettino della Unione matematica italiana
PY  - 2006
SP  - 299
EP  - 309
VL  - 9B
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_2_a3/
LA  - en
ID  - BUMI_2006_8_9B_2_a3
ER  - 
%0 Journal Article
%A Berti, Valeria
%T Existence and uniqueness for an integro-differential equation with singular kernel
%J Bollettino della Unione matematica italiana
%D 2006
%P 299-309
%V 9B
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_2_a3/
%G en
%F BUMI_2006_8_9B_2_a3
Berti, Valeria. Existence and uniqueness for an integro-differential equation with singular kernel. Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 2, pp. 299-309. http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_2_a3/

[1] M. Fabrizio - B. Lazzari, On the existence and asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rational Mech. Anal., 116 (2) (1991), 139-152. | Zbl

[2] M. Fabrizio - B. Lazzari, The domain of dependence inequality and asymptotic stability for a viscoelastic solid, Nonlinear Oscil., 1 (1998), 117-133. | Zbl

[3] M. Fabrizio - A. Morro, Mathematical problems in linear viscoelasticity, SIAM, Philadelphia, 1992. | Zbl

[4] G. Gentili, Regularity and stability for a viscoelastic material with a singular memory kernel, J. Elasticity, 37 (2) (1995), 139-156. | Zbl

[5] A. Hanyga, Wave propagation in media with singular memory, Math. Comput. Modelling, 34 (12-13) (2001), 1329-1421. | Zbl

[6] W.J. Hrusa - M. Renardy, On wave propagation in linear viscoelasticity, Quart. Appl. Math., 43 (2) (1985), 237-254. | Zbl

[7] O. A. Ladyzhenskaya, The boundary value problem of mathematical physics, Springer, New York, 1985. | Zbl

[8] M. Renardy - W. J. Hrusa - J. A. Nohel, Mathematical problems in viscoelasticity, Longman Scientific & Technical, John Wiley & Sons, New York, 1987.

[9] R. E. Showalter, Hilbert space methods for differential equations, Pitman, London, 1977. | Zbl

[10] F. Treves, Basic linear partial differential equations, Acad. press, New York, 1975. | Zbl