On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations

In this paper, the Harnack inequality result are established for a new class of the homogeneous nonlinear degenerate parabolic equations \begin{align*} div A(t,x,u,\nabla_x u)-\partial_t \vert u\vert^{p-2}u=0 \end{align*} on a bounded domain $D\subset R^{n+1}\mathrm{.}$ Let $A(t,x,\xi ,\eta )$ be measurable function on $R\times R^n\times R\times R^n\to R^n$ that satisfies the Caratheodory conditions for $\text{arbitrary }(t,x)\in D$ and $(\xi ,\eta )\in R^1\times R^n\mathrm{.}$ The following growth conditions are also satisfied: \begin{equation*} A(t,x,ξ,η)η\geq c_{1}ω(t,x)\vertη\vert^{p} \end{equation*} \begin{equation*} \vert A(t,x,ξ,η)\vert\leq c_{2}ω(t,x)\vertη\vert^{p-1},\quad p>1. \end{equation*} The exclusive Muckenhoupt condition ${\omega }^{\alpha }\in A_{1+\alpha \mathrm{/}r}\mathrm{.}$