Exponential attractors for abstract equations with memory and applications to viscoelasticity

Abstract

We consider an abstract equation with memory of the form $$\partial_t \boldsymbol{x}(t)+\int_{0}^\infty k(s) \boldsymbol{A}\boldsymbol{x}(t-s){\rm d} s+\boldsymbol{B}\boldsymbol{x}(t)=0$$ where $\boldsymbol{A},\boldsymbol{B}$ 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 $$\partial_{tt} u - h(0)\Delta u - \int_{0}^\infty h'(s) \Delta u(t-s){\rm d} s+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.