Singular perturbations in viscoelasticity (Q1335205)

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.