Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations

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 $\gamma >-1$, $F$ is uniformly $(\lambda ,\mathrm{\Lambda })$-elliptic with $0<\lambda \le \mathrm{\Lambda }$, $F(0)=0$, and $f\in C^0(Q_1)\cap L^{\mathrm{\infty }}(Q_1)$. Given any prescribed $\alpha \in (0,1)$, we prove that there exists $\epsilon =\epsilon (n,\alpha )>0$ such that $$\mathrm{\mid }\gamma \mathrm{\mid }+(\frac{\mathrm{\Lambda }}{\lambda }-1)\le \epsilon ⟹u\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\alpha ,\frac{1+\alpha }{2}}(Q_1)\mathrm{.}$$ In particular, the spatial gradient Hölder exponent can be chosen arbitrarily close to $1$ as $(\gamma ,\frac{\mathrm{\Lambda }}{\lambda })\to (0,1)$. This provides a fully nonlinear counterpart to the asymptotic gradient regularity result established in~\cite{WZJ26}.