On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations
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| Format: | Preprint |
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2023
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| _version_ | 1866913272699027456 |
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| author | Gasimov, Jasarat Mamedov, Farman |
| author_facet | Gasimov, Jasarat Mamedov, Farman |
| contents | 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} . $ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02026 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations Gasimov, Jasarat Mamedov, Farman Analysis of PDEs 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} . $ |
| title | On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.02026 |