  The axisymmetric kinematic dynamo problem is reconsidered and a number of
open questions are answered. Apart from axisymmetry and smoothness of data and
solution we deal with this problem under quite general conditions, i.e. we
assume a compressible fluid of variable (in space and time) conductivity moving
in an arbitrary (axisymmetric) domain. We prove unconditional, pointwise and
exponential decay of magnetic field and electric current to zero. The decay
rate of the external (meridional) magnetic field can become very small
(compared to free decay) for special flow fields and large magnetic Reynolds
numbers. We give an example of that. On the other hand, we show for fluids with
weak variation of mass density and conductivity that the meridional and
azimuthal decay rates do not drop significantly below those of free decay.
