Higher integrability for parabolic PDEs with generalized Orlicz growth

TL;DR

This paper establishes higher integrability results for solutions to nonlinear parabolic PDEs with generalized Orlicz growth, unifying and extending previous results across various growth conditions.

Contribution

It introduces a unified approach for higher integrability in parabolic PDEs with generalized Young function growth, covering known and new growth scenarios.

Findings

Proves higher integrability for a broad class of nonlinear parabolic systems.

Includes new borderline double phase and perturbed variable exponent growth cases.

Provides a simple proof of a reverse Hölder inequality applicable to singular and degenerate cases.

Abstract

We prove higher integrability of the gradient of weak solutions to nonlinear parabolic systems whose prototype is \[ \partial_t u-\mathrm{div}\Big(\frac{\varphi'(z, |\nabla u|)}{|\nabla u|}\nabla u\Big) =0, \qquad u=(u^1,\dots,u^N), \] where $\varphi$ is a generalized Young function. Special cases of our main theorem include previously known results for the $p$-growth, the variable exponent and the double phase growth. Also included are previously unknown borderline double phase growth and perturbed variable exponent growth, among others. The problem is controlled by a natural requirement of comparison of $\varphi$ between points in intrinsic parabolic cylinders via an (A1)-condition, which unifies disparate conditions from the special cases. Moreover, we handle both the singular and degenerate cases at the same time, providing a simple proof of a reverse H\"older type inequality, which is new even in the $p$-growth case.