  We consider the blow up problem in the energy space for the critical (gKdV)
equation in the continuation of part I and part II.
  We know from part I that the unique and stable blow up rate for solutions
close to the solitons with strong decay on the right is $1/t$. In this paper,
we construct non-generic blow up regimes in the energy space by considering
initial data with explicit slow decay on the right in space. We obtain finite
time blow up solutions with speed $t^{-\nu}$ where $ \nu>11/13,$ as well as
global in time growing up solutions with both exponential growth or power
growth. These solutions can be taken with initial data arbitrarily close to the
ground state solitary wave.
