Regularity for quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $${\mathrm{\partial }}_{t}u=[\mathrm{\mid }Du{\mathrm{\mid }}^q+a(x,t)\mathrm{\mid }Du{\mathrm{\mid }}^s](\mathrm{\Delta }u+(p-2)⟨D^{2}u\frac{Du}{\mathrm{\mid }Du\mathrm{\mid }},\frac{Du}{\mathrm{\mid }Du\mathrm{\mid }}⟩),$$ where $1<p<\mathrm{\infty }$, $-1<q\le s<\mathrm{\infty }$ and $a(x,t)\ge 0$. The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of $q,s$, such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when $q=p-2$ and $q<s$, it will encompass the parabolic $p$-Laplacian both in divergence form and in non-divergence form. We aim to explore the from $L^{\mathrm{\infty }}$ to $C^{1,\alpha }$ regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local H"{o}lder regularity of spatial gradients of viscosity solutions.