Evolution of a string network in backgrounds with rolling horizons

We discuss the temporal variation of the equation of state of a classical string network, evolving in a background in which the Hubble radius 1/H shrinks to a minimum and then re-expands to infinity. We also present a method to look for self-consistent non-vacuum string backgrounds, correponding to the simultaneous solution of the gravi-dilaton backgrounds equations and of the string equations of motion.