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We construct the Very Singular Solution (VSS) for the Anisotropic FastDiffusion Equation (AFDE) in the suitable good exponent range of fastdiffusion. VSS is a solution that, starting from an infinite mass located atone point as initial datum, evolves according to the corresponding equation asan admissible solution away from the singularity. It is expected to representimportant properties of the fundamental solutions when the initial mass is verybig. We work in the whole space.  In this setting we show that the diffusion process distributes mass from thethe initial infinite singularity along the different space directions: up toconstant factors, there is a simple partition formula for the anisotropic massexpansion, given approximately as the minimum of separate 1-D VSS solutions.This striking fact is a consequence of the improved scaling properties of thespecial solution, and it has strong consequences.  If we consider the family of Fundamental Solutions for different masses, weprove that they share the same universal tail behaviour as the VSS. Actually,their tail is asymptotically convergent to the unique VSS tail, so there is aVSS partition formula for their profile at spatial infinity.  With the help of this analysis we study the behaviour of the class ofnonnegative finite-mass solutions of the Anisotropic FDE, and prove the GlobalHarnack Principle (GHP) and the Asymptotic Convergence in Relative Error (ACRE)under a natural assumption on the decay of the initial tail.

The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences