On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations

Abstract

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}. $ 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 $(\xi,\eta)\in R^{1}\times R^n.$ The following growth conditions are also satisfied: \begin{equation*} A(t,x,\xi,\eta)\eta\geq c_{1}\omega(t,x)\vert\eta\vert^{p} \end{equation*} \begin{equation*} \vert A(t,x,\xi,\eta)\vert\leq c_{2}\omega(t,x)\vert\eta\vert^{p-1},\quad p>1. \end{equation*} The exclusive Muckenhoupt condition $ \omega^{\alpha} \in A_{1+{\alpha}/r} . $