Regularity estimates to quasi-linear parabolic equations in non-divergence form with non-homogeneous signature

In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this class is given by $${\mathrm{\partial }}_{t}u=(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{\mathfrak{p}}+\mathfrak{a}(x,t)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{\mathfrak{q}}){\mathrm{\Delta }}_p^{\mathrm{N}}u+f(x,t)\text{in}{Q}_1=B_1\times (-1,0],$$ where $p\in (1,\mathrm{\infty })$, $\mathfrak{p},\mathfrak{q}\in [0,\mathrm{\infty })$, and $\mathfrak{a},f:Q_1\to \mathbb{R}$ are suitably defined functions. Our approach is based on geometric tangential methods, incorporating a refined oscillation mechanism, compactness arguments, ``alternative methods,‘’ and scaling techniques. Furthermore, we derive pointwise estimates in settings exhibiting singular-degenerate or doubly singular signatures. To some extent, our regularity estimates refine and extend previous results from \cite{FZ23} through distinct methodological advancements. Finally, we explore connections between our findings and fundamental nonlinear models in the theory of quasilinear PDEs, which may be of independent interest.