  Axion scenarios in which the spontaneous breaking of the Peccei-Quinn
symmetry takes place before or during inflation, and in which axion dark matter
arises from the misalignment mechanism, can be constrained by Cosmic Microwave
Background isocurvature bounds. Dark matter isocurvature is thought to be
suppressed in models with axion-inflaton interactions, for which axion
perturbations are assumed to freeze at horizon crossing during inflation.
However, this assumption can be an oversimplification due to the interactions
themselves. In particular, non-perturbative effects during reheating may lead
to a dramatic growth of axion perturbations. We perform lattice calculations in
two models in which the Peccei-Quinn field participates in inflation. We find
that the growth of axion perturbations is such that the Peccei-Quinn symmetry
is restored for an axion decay constant $f_A\lesssim10^{16}$-$10^{17}$ GeV,
leading to an over-abundance of dark matter, unless $f_A \lesssim 2 \times
10^{11}$ GeV. For $f_A\gtrsim10^{16}$-$10^{17}$ GeV we still find a large
growth of axion perturbations at low momentum, such that a naive extrapolation
to CMB scales suggests a violation of the isocurvature bounds.
