×

zbMATH — the first resource for mathematics

Singular perturbations in viscoelasticity. (English) Zbl 0805.73029
We study the singular perturbation for a class of partial integro- differential equations in viscoelasticity of the form \[ \rho u^ \rho_{tt} (t,x) = Eu^ \rho_{xx} (t,x) + \int ^ t _{-\infty} a (t-s) u^ \rho_{xx} (s,x) ds + \rho g (t,x) + f (x),\tag{a} \] when the density \(\rho\) of the material goes to zero. We will prove that when \(\rho \to 0\) the solutions of the dynamical systems (a) (with \(\rho > 0)\) approach the solution of the steady state obtained from equation (a) with \(\rho=0\). The technique of energy estimates is used.

MSC:
74Hxx Dynamical problems in solid mechanics
45K05 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C.M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity , J. Differential Equations 7 (1970), 544-569. · Zbl 0212.45302 · doi:10.1016/0022-0396(70)90101-4
[2] J. Dieudonne, Foundations of modern analysis , Academic Press, New York, 1969.
[3] C.M. Dafermos and J.A. Nohel, A nonlinear hyperbolic Volterra equation in viscoelasticity , Amer. J. Math. Suppl. (1981), 87-116. · Zbl 0588.35016
[4] ——–, A nonlinear hyperbolic Volterra equation in viscoelasticity, Contributions to analysis and geometry , The Johns Hopkins University Press, 1981, 87-116. · Zbl 0588.35016
[5] H.O. Fattorini, Second order linear differential equations in Banach spaces , North-Holland, 1985, 165-222. · Zbl 0564.34063
[6] R. Grimmer and J.H. Liu, Integrated semigroups and integrodifferential equations , to appear in Semigroup Forum. · Zbl 0788.45011 · doi:10.1007/BF02573656 · eudml:135299
[7] ——–, Integrodifferential equations with non-densely defined operators , in Differential equations with appplications in biology, physics and engineering , J. Goldstein, F. Kappel and W. Schappacher (eds.), Marcel Dekker, New York, 1991, 185-199.
[8] R. Grimmer and E. Sinestrari, Maximum norm in one-dimensional hyperbolic problems , J. Differential Integral Equations 5 (1992), 421-432. · Zbl 0782.47037
[9] M.L. Heard, A class of hyperbolic Volterra integrodifferential equations , Nonlinear Anal. 8 (1984), 79-93. · Zbl 0535.45007 · doi:10.1016/0362-546X(84)90029-4
[10] J. Hudson, The excitation and propagation of elastic waves , Cambridge University Press, 1980, 188-219. · Zbl 0435.35002
[11] W.J. Hrusa and M. Renardy, On a class of quasilinear partial integrodifferential equations with singular kernels , J. Differential Equations 64 (1986), 195-220. · Zbl 0593.45011 · doi:10.1016/0022-0396(86)90087-2
[12] V. Lakshmikantham and S. Leela, Differential and integral inequalities , Academic Press, New York, 1969. · Zbl 0177.12403
[13] R.C. MacCamy, An integro-differential equation with application in heat flow , Quart. Appl. Math. 35 (1977), 1-19. · Zbl 0351.45018
[14] ——–, A model for one-dimensional, nonlinear viscoelasticity , Quart. Appl. Math. 35 (1977), 21-33. · Zbl 0355.73041
[15] ——–, Approximations for a class of functional differential equtions , SIAM J. Appl. Math. 23 (1972), 70-83. JSTOR: · Zbl 0237.34102 · doi:10.1137/0123008 · links.jstor.org
[16] R.K. Miller, Nonlinear Volterra integral equations , W.A. Benjamin Inc., 1971, 189-233. · Zbl 0448.45004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.