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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
##### Keywords:
dynamical systems; energy estimates
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##### References:
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