  We study axisymmetric solution to the conformally invariant wave equation on
a Kerr background by means of numerical and analytical methods. Our main focus
is on the behaviour of the solutions near spacelike infinity, which is
appropriately represented as a cylinder. Earlier studies of the wave equation
on a Schwarzschild background have revealed important details about the
regularity of the corresponding solutions. It was found that, on the cylinder,
the solutions generically develop logarithmic singularities at infinitely many
orders. Moreover, these singularities also `spread' to future null infinity.
However, by imposing certain regularity conditions on the initial data, the
lowest-order singularities can be removed. Here we are interested in a
generalisation of these results to a rotating black hole background and study
the influence of the rotation rate on the properties of the solutions. To this
aim, we first construct a conformal compactification of the Kerr solution which
yields a suitable representation of the cylinder at spatial infinity. Besides
analytical investigations on the cylinder, we numerically solve the wave
equation with a fully pseudospectral method, which allows us to obtain highly
accurate numerical solutions. This is crucial for a detailed analysis of the
regularity of the solutions. In the Schwarzschild case, the numerical problem
could effectively be reduced to solving $(1+1)$-dimensional equations. Here we
present a code that can perform the full $2+1$ evolution as required for
axisymmetric waves on a Kerr background.
