Higher integrability for parabolic PDEs with generalized Orlicz growth
Peter Hästö, Jihoon Ok
TL;DR
This work addresses the question of higher integrability for the gradient of weak solutions to parabolic systems with generalized Orlicz growth. It introduces an intrinsic parabolic (A1) condition on the growth function $\varphi$ and develops a mollification-based, unified framework that covers singular, degenerate, and intermediate growth regimes. The analysis yields a reverse Hölder-type inequality and, via a Vitali-type covering and stopping-time argument, proves that $\varphi(\cdot,|\nabla u|)\in L^{1+\varepsilon}_{\text{loc}}$ for some $\varepsilon>0$, with explicit dependence on local averages through the operator $$(\mathcal{D}^-)^{-1}$$ applied to $\fint_{Q_{2r}}[\varphi(z,|\nabla u|)+1]dz$. This framework recovers known results for $p$-growth, variable exponent, and double-phase growth, and also handles borderline/perturbed variants, offering a simpler, unified approach that does not rely on Lipschitz truncation. Overall, the paper advances the regularity theory for nonlinear parabolic systems with nonstandard growth by providing a robust, intrinsic method that extends classical results to a broad class of growth conditions.
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 Hölder type inequality, which is new even in the $p$-growth case.