  Evolution of inhomogeneities in the axion field around the QCD epoch is
studied numerically, including for the first time important non-linear effects.
It is found that perturbations on scales corresponding to causally disconnected
regions at $T \sim 1 \, {\rm GeV}$ can lead to very dense axion clumps, with
present density $\rho_a \ga 10^{-8}\,{\rm g \, cm^{-3}}$. This is high enough
for the collisional $2a \rightarrow 2a$ process to lead to Bose--Einstein
relaxation in the gravitationally bound clumps of axions, forming Bose stars.
