Self-improving property of the fast diffusion equation

TL;DR

This paper proves local higher integrability of the gradient of solutions to the fast diffusion equation, extending existing theory for the singular case using intrinsic metrics and Calderón-Zygmund techniques.

Contribution

It introduces a reverse Hölder inequality and an intrinsic Calderón-Zygmund covering argument for the fast diffusion equation in the singular regime, extending prior results for m≥1.

Findings

Established reverse Hölder inequality for the gradient

Proved local higher integrability of the gradient

Extended theory to the singular case m in ((n-2)_+/n+2,1)

Abstract

We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse H\"older inequality in suitable intrinsic cylinders. Relying on an intrinsic Calder\'on-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $m\in\left(\frac{(n-2)_+}{n+2},1\right)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $m\geq 1$ (see [GS16] in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.