Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations

Abstract

This is a continuation of our recent work~\cite{WZJ26} on asymptotic gradient regularity for nonlinear parabolic equations. In this paper, we consider bounded viscosity solutions of the degenerate or singular fully nonlinear parabolic equation \begin{equation*} u_t-|Du|^γF(D^2u)=f \quad \text{in } Q_1, \end{equation*} where $ γ>-1 $, $ F $ is uniformly $ (λ, Λ)$-elliptic with $0<λ\leqΛ$, $F(0)=0$, and $f\in C^0(Q_1)\cap L^\infty(Q_1)$. Given any prescribed $α\in(0,1)$, we prove that there exists $ \varepsilon=\varepsilon(n,α)>0 $ such that \[ |γ|+\left(\fracΛλ-1\right) \leq \varepsilon \quad \Longrightarrow \quad u \in C_{\mathrm{loc}}^{1+α,\frac{1+α}{2}}(Q_1). \] In particular, the spatial gradient Hölder exponent can be chosen arbitrarily close to $1$ as $ (γ, \fracΛλ) \to (0,1) $. This provides a fully nonlinear counterpart to the asymptotic gradient regularity result established in~\cite{WZJ26}.