  The Bramson logarithmic shift of the position of pulled fronts is a universal
feature common to a large class of monostable traveling wave equations. As one
varies the non-linearities it so happens that one can observe, at some critical
non linearity, a transition from pulled fronts to pushed fronts. At this
transition the Bramson shift is modified. In the limit where time goes to
infinity and the non-linearity becomes critical, the position of the front
exhibits a cross-over. The goal of the present note is to give the expression
of this cross-over function, for a particular model which is exactly soluble,
with the hope that this expression would remain valid for more general
traveling wave equations at the transition between pulled and pushed fronts.
Other cross-over functions are also obtained, for this particular model, to
describe the dependence on initial conditions or the effect of a cut-off.
