the space and the approximation on which our Kähler potential  is based brakes down. In particular this means an evolution towards a state with small blow-ups at late time, $y\rightarrow\infty$ is not possible.

Potential with a single exponential
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Let us now analyse the solutions more systematically, starting from simple patterns of evolution of the blow-up moduli and moving to more complicated ones.

Certainly, the simplest possibility is to freeze all blow-up moduli by setting all $\z_i=0$ in Eq. . In this case, the effective potential  for the bulk moduli vanishes identically and the equations of motion  for $\f_A$ can be easily integrated. In proper time $t$, related to $\t$ by Eq. , one easily finds as the general solution in this case $$\f_A = q_A\ln\left(\frac{|t-t_0|}{T}\right)+k_A\label{frees}$$ where $t_0$, $T$ and $k_A$ are constants. The expansion powers $q_A$ satisfy $$|{\bf q}|^2=\frac{4}{3}\; ,$$ which follows from the constraint , and are otherwise arbitrary. These are simply solutions describing power-law evolution of the bulk moduli, which, in fact, are identical to the ones that can be obtained for M-theory on a seven-dimensional torus.

We now move to the next more complicated case where blow-up moduli of only one particular type $(\t ,a)$ evolve 