Self-improving property of degenerate parabolic equations of porous medium-type

TL;DR

This paper proves that solutions to certain degenerate parabolic equations exhibit higher integrability of their gradients by establishing a reverse Hölder inequality using an intrinsic Calderón-Zygmund covering argument.

Contribution

It introduces a novel intrinsic Calderón-Zygmund covering method to demonstrate local higher integrability of gradients for porous medium-type equations.

Findings

Gradient of solutions satisfies reverse Hölder inequality

Established local higher integrability of solution gradients

Modified classical Gehring lemma with intrinsic covering argument

Abstract

We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse H\"older inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calder\'on-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution $u$.