Before revision:   An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin.
After revision:  The spatial part of the degenerate diffusion coefficient is a smooth function of r and z independent of the solution and vanishes like O(r^2)+O(z^2) near the origin. This potential singularity is induced by a potential singularity of the 3D Euler equations. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin.