Exponential Attractors For Abstract Equations With Memory And Applications To Viscoelasticity

Abstract

We consider an abstract equation with memory of the form partial derivative(t)x(t) + integral(infinity)(0) k(s)Ax(t-s)ds + Bx(t) = 0 where A, B are operators acting on some Banach space, and the convolution kernel k is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation partial derivative(tt)u - h(0)Delta u - integral(infinity)(0) h'(s)Delta u(t-s)ds + f(u) = g arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel h(s) = k(s) + 1.