Higher order Schauder estimates for degenerate or singular parabolic equations

Abstract

In this paper, we complete the analysis initiated in [Calc. Var. Partial Differential Equations 63 (2024), article no. 204] establishing some higher order $C^{k+2,\alpha }$ Schauder estimates ($k\in \mathbb{N}$) for a class of parabolic equations with weights that are degenerate/singular on a characteristic hyperplane. The $C^{2,\alpha }$-estimates are obtained through a blow-up argument and a Liouville theorem, while the higher order estimates are obtained by a fine iteration procedure. As a byproduct, we present two applications. First, we prove similar Schauder estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type. Second, we provide an alternative proof of the higher order boundary Harnack principles established in [J. Differential Equations 260 (2016), 1801–1829] and [Discrete Contin. Dyn. Syst. 42 (2022), 2667–2698].