Higher integrability for parabolic PDEs with generalized Orlicz growth

Abstract

We prove higher integrability of the gradient of weak solutions to nonlinear parabolic systems whose prototype is \[ \partial_t u-\mathrm{div}\Big(\frac{φ'(z, |\nabla u|)}{|\nabla u|}\nabla u\Big) =0, \qquad u=(u^1,\dots,u^N), \] where $φ$ 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 $φ$ 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 unified proof of of a reverse Hölder type inequality.