  We study cosmological implications of the duality ($PSL(2,{\bf Z})$)
invariant potential for the compactification radius $T$, arising in a class of
superstring vacua. We show that in spite of having only one minimum in the
fundamental domain of the $T$ field there are two types of non-supersymmetric
domain walls: one is associated with the discrete Peccei-Quinn symmetry $T\to
T+i$, analogous to the axionic domain wall, and another one associated with the
noncompact symmetry $T\to 1/T$, analogous to the $Z_2$ domain walls. The first
one is bound by stringy cosmic strings. The scale of such domain walls is
governed by the scale of gaugino condensation (${\cal O} (10^{16}$ GeV) in the
case of hidden $E_8$ gauge group), while the separation between minima is of
order $M_{pl}$. We discuss the formation of walls and their cosmological
implications: the walls must be gotten rid of, either by chopping by stringy
cosmic strings and/or inflation. Since there is no usual Kibble mechanism to
create strings, either one must assume they exist $ab initio$, or one must
conclude that string cosmologies require inflation. The non-perturbative
potential dealt with here appears not to give the needed inflationary epoch.
