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On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations

Jasarat Gasimov, Farman Mamedov

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 $A(t,x,ξ,η)$ 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 $(ξ,η)\in R^{1}\times R^n.$ 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 $ ω^α \in A_{1+α/r} . $

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 Let be measurable function on that satisfies the Caratheodory conditions for and 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
Paper Structure (4 sections, 7 theorems, 122 equations)

This paper contains 4 sections, 7 theorems, 122 equations.

Table of Contents

  1. Introduction.
  2. Preliminary
  3. Moser's inequality
  4. Harnack's inequality

Key Result

Corollary 3

FM Let $q\in \left [ 1, (n+p)/n \right ].$ Then for a function $f\in Lip_0(D)$ the the inequality holds, where $A=\vert D \vert ^{\frac{1}{n}-\frac{q-1}{p}}, \, \, C_0$ depends only on $n, q.$

Theorems & Definitions (12)

  • Definition 1
  • Corollary 3
  • Lemma 4
  • Corollary 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • Lemma 9
  • ...and 2 more