On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations

TL;DR

This paper establishes a Harnack inequality for a new class of homogeneous nonlinear degenerate parabolic equations, expanding understanding of their regularity properties under specific growth and weight conditions.

Contribution

The paper introduces a Harnack inequality for a novel class of degenerate parabolic equations with weighted growth conditions, extending previous results to more general nonlinear and degenerate cases.

Findings

Harnack inequality proven for the new class of equations

Results applicable under Muckenhoupt weight conditions

Enhanced understanding of regularity for 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} . $