Exponential attractors for abstract equations with memory and applications to viscoelasticity

Abstract

We consider an abstract equation with memory of the form $${\partial }_t\mathbf{x}(t)+{\int }_0^{\infty }k(s)\mathbf{A}\mathbf{x}(t-s)\mathrm{d}s+\mathbf{B}\mathbf{x}(t)=0$$ where $\mathbf{A},\mathbf{B}$ 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 $${\partial }_{tt}u-h(0)\Delta u-{\int }_0^{\infty }{h}^{\prime }(s)\Delta u(t-s)\mathrm{d}s+f(u)=g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.