Attractors for Semi-Linear Equations of Viscoelasticity with Very Low Dissipation

Abstract

We analyze a differential system arising in the theory of isothermal viscoelasticity. This system is equivalent to an integrodifferential equation of hyperbolic type with a cubic nonlinearity, where the dissipation mechanism is contained only in the convolution integral, accounting for the past history of the displacement. In particular, we consider here a convolution kernelwhich entails an extremely weak dissipation. In spite of that,we show that the related dynamical system possesses a globalattractor of optimal regularity.