Unified approach to $C^{1,\alpha}$ regularity for quasilinear parabolic equations with applications

In this paper, we are interested in obtaining a unified approach for $C^{1,\alpha}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman. As an application of these techniques, we obtain $C^{1,\alpha}$ regularity (with sharp assumptions) for two parabolic non-standard growth problems.