  In the framework of superstring compactifications with N=1 supersymmetry
spontaneously broken, (by either geometrical fluxes, branes or else), we show
the existence of new inflationary solutions. The time-trajectory of the scale
factor of the metric a, the supersymmetry breaking scale m=m(Phi) and the
temperature T are such that am and aT remain constant. These solutions request
the presence of special moduli-fields: i) The universal "no-scale-modulus" Phi,
which appears in all N=1 effective supergravity theories and defines the
supersymmetry breaking scale m(Phi). ii) The modulus Phi_s, which appears in a
very large class of string compactifications and has a Phi-dependent kinetic
term. During the time evolution, a^4 rho_s remains constant as well, (rho_s
being the energy density induced by the motion of Phi_s). The cosmological term
Lambda(am), the curvature term k(am, aT) and the radiation term c_R=a^4 rho are
dynamically generated in a controllable way by radiative and temperature
corrections; they are effectively constant during the time evolution. Depending
on Lambda, k and c_R, either a first or second order phase transition can occur
in the cosmological scenario. In the first case, an instantonic Euclidean
solution exists and connects via tunneling the inflationary evolution to
another cosmological branch. The latter starts with a big bang and, in the case
the transition does not occur, ends with a big crunch. In the second case, the
big bang and the inflationary phase are smoothly connected.
