Schauder estimates for quasilinear parabolic equations with general Orlicz growth

Abstract

We develop a unified approach to Schauder estimates for scalar parabolic equations \[ \partial_t u-\operatorname{div}\big(\mathcal{A}(x,t,Du)\big)=f(x,t) \] with general Orlicz growth and coefficients that are H\"older continuous in the spatial variables and merely measurable in time. Combining intrinsic gradient estimates with a perturbative argument, we establish quantitative local bounds and H\"older estimates for the spatial gradient in the supercritical range and, for locally bounded weak solutions, throughout the full admissible growth range. Our results extend the Schauder regularity theory for parabolic $p$-Laplace type equations to a broad class of possibly nonradial operators with Orlicz growth, while simplifying the regularity arguments and treating singular, degenerate, and mixed growth laws within a unified framework.