  Axion-like particles (ALPs) are often considered as good candidates for dark
matter. Several mechanisms generating relic abundance of ALP dark matter have
been proposed. They may involve processes which take place before, during or
after cosmic inflation. In all cases an important role is played by the
potential of the corresponding Peccei-Quinn (PQ) field. Quite often this
potential is assumed to be dominated by a quartic term with a very small
coupling. We show that in such situation it is crucial to take into account
different kinds of corrections especially in models in which the PQ field
evolves during and after inflation. We investigate how such evolution changes
due to radiative, thermal and geometric corrections. In many cases those
changes are very important and result in strong modifications of the
predictions of a model. They may strongly influence the amount of ALP
contributions to cold and warm components of dark matter as well as the power
spectrum of associated isocurvature perturbations. Models with a
quasi-supersymmetric spectrum of particles to which the PQ field couples seem
to be especially interesting. Qualitative features of such models are discussed
with the help of approximate analytical formulae. However, the dynamics of the
PQ field with the considered corrections taken into account is more complicated
than in the case without corrections so dedicated numerical calculations are
necessary to obtain precise predictions. We present such results for some
characteristic benchmark points in the parameter space.
