  This paper focuses on propagation phenomena in reaction-diffusion equations
with a weaklymonostable nonlinearity. The reaction term can be seen as an
intermediate between the classicallogistic one (or Fisher-KPP) and the standard
weak Allee effect one. We investigate the effect ofthe decay rate of the
initial data on the propagation rate. When the right tail of the initial datais
sub-exponential, finite speed propagation and acceleration may happen and we
derive the exactseparation between the two situations. When the initial data is
sub-exponentially unbounded, accel-eration unconditionally occurs. Estimates
for the locations of the level sets are expressed in termsof the decay of the
initial data. In addition, sharp exponents of acceleration for initial data
withsub-exponential and algebraic tails are given. Numerical simulations are
presented to illustrate theabove findings.
