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Spectral properties of a type of integro-differential stiff problems. (English) Zbl 0566.45015
The author studies an integro-differential stiff problem in \(R^ n\), which models the vibrations of a linear viscoelastic body. The existence and uniqueness of the solution is obtained by reformulating the problem in a Hilbert space setting and showing that the corresponding operator is the infinitesimal generator of a contraction semigroup. A formal asymptotic expansion is then constructed, which converges to the solution under certain regularity conditions. Also, it is shown that the eigenvalues of two operators related to the operator of the original stiff problem are accumulation points of the spectrum of the latter.
Reviewer: C.Constanda

45K05 Integro-partial differential equations
47D03 Groups and semigroups of linear operators
74Hxx Dynamical problems in solid mechanics
Full Text: DOI EuDML
[1] H. BREZIS, Opérateurs maximaux monotones, North-Holland, Amsterdam (1973).
[2] C. M. DAFERMOS, An abstract Volterra equation with application to linear viscoelasticity, J. Diff. Équations, 7 (1970), pp. 554-569. Zbl0212.45302 MR259670 · Zbl 0212.45302 · doi:10.1016/0022-0396(70)90101-4
[3] C. M. DAFFERMOS, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal., 37 (1970), pp. 297-308. Zbl0214.24503 MR281400 · Zbl 0214.24503 · doi:10.1007/BF00251609
[4] C. M. DAFERMOS, Contraction semigroups and trend to equilibrium in continuum mechanics, Lec. Notes Math., 503, Springer, Berlin (1975), pp. 295-306. Zbl0345.47032 · Zbl 0345.47032
[5] J. L. LIONS, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lec. Notes Math., 323, Springer, Berlin (1973). Zbl0268.49001 MR600331 · Zbl 0268.49001
[6] M. LOBO-HIDALGO, Propriétés spectrales de certaines équations différentielles intervenant en viscoélasticité, Rend. Sem. Mat. Univ. Polit. Torino, 39 (1981), pp. 33-51. Zbl0489.73063 MR660992 · Zbl 0489.73063
[7] M. LOBO-HIDALGO and E. SANCHEZ-PALENCIA, Perturbation of spectral properties for a class of stiff problems, Proc. Fourth Inter. Symp. Comp. Method. in Science and Engineering, North-Holland, Amsterdam (1980). Zbl0448.35078 MR584063 · Zbl 0448.35078
[8] V. PETERSON, V. V. VARADAN and Y. K. VARADAN, Scattering of acoustic waves by layered elastic and viscoelastic obstacles in water, J. Acoust. Soc. Am., 68 (1980), pp. 673-685. Zbl0465.76077 MR579191 · Zbl 0465.76077 · doi:10.1121/1.384726
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