Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917696145195008 |
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| author | Iagar, Razvan Gabriel Laurençot, Philippe |
| author_facet | Iagar, Razvan Gabriel Laurençot, Philippe |
| contents | Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $$\partial_t u - \mathrm{div}(|\nabla u|^{p-2}\nabla u) +|\nabla u|^{q}=0 \qquad {\rm in} \ (0,\infty)\times\mathbb{R}^N$$ are studied when $N\geq1$ and the exponents $(p,q)$ satisfy $p_c=\frac{2N}{N+1}$, $p-1<q<\frac{p}{2}$. Existence and uniqueness of such a solution are established in dimension $N=1$. In dimension $N\geq2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as $|x|\to\infty$ is provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11518 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption Iagar, Razvan Gabriel Laurençot, Philippe Analysis of PDEs Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $$\partial_t u - \mathrm{div}(|\nabla u|^{p-2}\nabla u) +|\nabla u|^{q}=0 \qquad {\rm in} \ (0,\infty)\times\mathbb{R}^N$$ are studied when $N\geq1$ and the exponents $(p,q)$ satisfy $p_c=\frac{2N}{N+1}$, $p-1<q<\frac{p}{2}$. Existence and uniqueness of such a solution are established in dimension $N=1$. In dimension $N\geq2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as $|x|\to\infty$ is provided. |
| title | Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.11518 |