We investigate (n+1)-dimensional string-dilaton cosmology with effective dilation potential in presence of perfect-fluid matter. We get exact solutions parametrized by the constant #gamma# of the state equation p = (#gamma# - 1)#rho#, the spatial dimension number n, the bulk of matter, and the spatial curvature constant #kappa#. Several interesting cosmological behaviours are selected. Finally we discuss the recovering of ordinary Einstein gravity starting from string dominated regime and a sort of asymptotic freedom due to string effective coupling.
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