  We look at the possibility of superinflationary behavior in a class of
anisotropic Type-IIB superstring cosmologies in the context of Pre-big Bang
scenario and find that there exists a rather narrow range of parameters for
which these models inflate. We then show that, although in general this
behavior is left untouched by the introduction of a Ramond-Ramond axion field
through a SL(2,R) rotation, there exists a particular class of axions for which
inflation disappears completely. Asymptotic past initial conditions are briefly
discussed, and some speculations on the possible extension of Pre-big Bang
ideas to gravitational collapse are presented.
