  We study an anisotropic, possibly non-homogeneous version of the evolution
$p$-Laplacian equation when fast diffusion holds in all directions. We develop
the basic theory and prove symmetrization results from which we derive $L^1$ to
$L^\infty$ estimates. We prove the existence of a self-similar fundamental
solution of this equation in the appropriate exponent range, and uniqueness in
a smaller range. We also obtain the asymptotic behaviour of finite mass
solutions in terms of the self-similar solution. Positivity, decay rates as
well as other properties of the solutions are derived. The combination of
self-similarity and anisotropy is not common in the related literature. It is
however essential in our analysis and creates mathematical difficulties that
are solved for fast diffusions.
