A new approach to equations with memory

We discuss a novel approach to the mathematical analysis of equations with memory, based on the notion of a {\it state}. This is the initial configuration of the system at time $t=0$ which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function $G:\R^+\to\R^+$ such that
$$G(0)=\lim_{s\to 0}G(s)>\lim_{s\to\infty}G(s)>0$$
we consider an abstract version of the evolution equation
$$\partial_{tt} {\bm u}({\bm x},t)-\Delta\Big[G(0) {\bm u}({\bm x},t) +\int_0^\infty G'(s) {\bm u}({\bm x},t-s)
{\rm d} s\Big]=0$$
arising from linear viscoelasticity.