  Starting from an initial classical four dimensional flat background of the
heterotic or type II superstrings, we are able to determine at the string
one-loop level the quantum corrections to the effective potential due to the
spontaneous breaking of supersymmetry by "geometrical fluxes". Furthermore,
considering a gas of strings at finite temperature, the full "effective thermal
potential" is determined, giving rise to an effective non-trivial pressure. The
backreaction of the quantum and thermal corrections to the space-time metric as
well as to the moduli fields induces a cosmological evolution that depends on
the early time initial conditions and the number of spontaneously broken
supersymmetries. We show that for a whole set of initial conditions, the
cosmological solutions converge at late times to two qualitatively different
trajectories: They are either attracted to (i) a thermal evolution similar to a
radiation dominated cosmology, implemented by a coherent motion of some moduli
fields, or to (ii) a "Big Crunch" non-thermal cosmological evolution dominated
by the non-thermal part of the effective potential or the moduli kinetic
energy. During the attraction to the radiation-like era, periods of accelerated
cosmology can occur. However, they do not give rise to enough inflation (e-fold
< 0.2) for the models we consider, where N>1 supersymmetry is spontaneously
broken to N=0.
