  We study the pairwise annihilation process $A+A\to$ inert of a number of
random walkers, which originally are localized in a small region in space. The
size of the colony and the typical distance between particles increases with
time and, consequently, the reaction rate goes down. In the long time limit the
spatial density profile becomes scale invariant. The mean-field approximation
of this scenario bears some surprises. It predicts an upper critical dimension
$d_c=2$, with logarithmic corrections at the critical dimension and nontrivial
scaling behavior for d<2. Based on an exact solution of the one dimensional
system we conjecture that the mean-field {\it exponents} are in fact correct
even below the upper critical dimension, down to d=1, while the corresponding
scaling function that describes the spatial density profile, changes for d<2
due to the stochastic fluctuations.
