Degenerate parabolic $p$-Laplacian equations: existence, uniqueness and asymptotic behavior of solutions
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| Format: | Preprint |
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2024
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| _version_ | 1866916915228704768 |
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| author | Cruz-Uribe, David Moen, Kabe Shao, Yuanzhen |
| author_facet | Cruz-Uribe, David Moen, Kabe Shao, Yuanzhen |
| contents | In this paper we study the degenerate parabolic $p$-Laplacian,$ \partial_t u - v^{-1}{\rm div}(|\sqrt{Q} \nabla u|^{p-2} Q \nabla u)=0$, where the degeneracy is controlled by a matrix $Q$ and a weight $v$. With mild integrability assumptions on $Q$ and $v$, we prove the existence and uniqueness of solutions on any interval $[0,T]$. If we further assume the existence of a degenerate Sobolev inequality with gain, the degeneracy again controlled by $v$ and $Q$, then we can prove both finite time extinction and ultracontractive bounds. Moreover, we show that there is equivalence between the existence of ultracontractive bounds and the weighted Sobolev inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_13849 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degenerate parabolic $p$-Laplacian equations: existence, uniqueness and asymptotic behavior of solutions Cruz-Uribe, David Moen, Kabe Shao, Yuanzhen Analysis of PDEs Primary: 35K65, 46E35, Secondary: 35D30, 42B35, 42B37 In this paper we study the degenerate parabolic $p$-Laplacian,$ \partial_t u - v^{-1}{\rm div}(|\sqrt{Q} \nabla u|^{p-2} Q \nabla u)=0$, where the degeneracy is controlled by a matrix $Q$ and a weight $v$. With mild integrability assumptions on $Q$ and $v$, we prove the existence and uniqueness of solutions on any interval $[0,T]$. If we further assume the existence of a degenerate Sobolev inequality with gain, the degeneracy again controlled by $v$ and $Q$, then we can prove both finite time extinction and ultracontractive bounds. Moreover, we show that there is equivalence between the existence of ultracontractive bounds and the weighted Sobolev inequality. |
| title | Degenerate parabolic $p$-Laplacian equations: existence, uniqueness and asymptotic behavior of solutions |
| topic | Analysis of PDEs Primary: 35K65, 46E35, Secondary: 35D30, 42B35, 42B37 |
| url | https://arxiv.org/abs/2405.13849 |