Higher integrability for parabolic PDEs with generalized Orlicz growth

We prove higher integrability of the gradient of weak solutions to nonlinear parabolic systems whose prototype is $${\mathrm{\partial }}_{t}u-\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{{\phi }^{\mathrm{\prime }}(z,\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })}{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }}\mathrm{\nabla }u\right)=0,u=(u^1,\dots ,u^N),$$ where $\phi$ 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 $\phi$ 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.