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

Abstract

We construct the Very Singular Solution (VSS) for the Anisotropic Fast Diffusion Equation (AFDE) in the suitable good exponent range of fast diffusion. VSS is a solution that, starting from an infinite mass located at one point as initial datum, evolves according to the corresponding equation as an admissible solution away from the singularity. It is expected to represent important properties of the fundamental solutions when the initial mass is very big. We work in the whole space. In this setting we show that the diffusion process distributes mass from the the initial infinite singularity along the different space directions: up to constant factors, there is a simple partition formula for the anisotropic mass expansion, given approximately as the minimum of separate 1-D VSS solutions. This striking fact is a consequence of the improved scaling properties of the special solution, and it has strong consequences. If we consider the family of Fundamental Solutions for different masses, we prove 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 a VSS partition formula for their profile at spatial infinity. With the help of this analysis we study the behaviour of the class of nonnegative finite-mass solutions of the Anisotropic FDE, and prove the Global Harnack Principle (GHP) and the Asymptotic Convergence in Relative Error (ACRE) under a natural assumption on the decay of the initial tail.