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 suitably good exponent range. VSS is a solution that, starting from an infinite mass located at one point as initial datum, evolves as an admissible solution of the corresponding equation everywhere away from the point singularity. It is expected to represent important properties of the fundamental solutions when the initial mass is very big. Here we work in the whole Euclidean space.

In this setting we show how the diffusion process distributes mass from the initial infinite singularity with different rates along the different space directions. Indeed, and 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 all share the same universal tail behaviour (i.e., for large $\left|x\right|$) as the VSS. Namely, their tail is asymptotically convergent to the unique VSS tail. This means that the VSS partition formula holds also for the fundamental solutions 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.