Self-improving property of the fast diffusion equation

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.