Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910555633090560 |
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| author | Iagar, Razvan Gabriel Laurençot, Philippe Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Laurençot, Philippe Sánchez, Ariel |
| contents | Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q = 0 \;\;\text{ in }\;\; (0,\infty)\times\mathbb{R}^N,$$ with $m>1$, $q\in(0,1)$ and $σ=2(1-q)/(m-1)$. Looking for solutions of the form $$ u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$ it is shown that, for $m+q>2$, there exists a unique exponent $β_*\in(0,\infty)$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles Iagar, Razvan Gabriel Laurençot, Philippe Sánchez, Ariel Analysis of PDEs Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q = 0 \;\;\text{ in }\;\; (0,\infty)\times\mathbb{R}^N,$$ with $m>1$, $q\in(0,1)$ and $σ=2(1-q)/(m-1)$. Looking for solutions of the form $$ u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$ it is shown that, for $m+q>2$, there exists a unique exponent $β_*\in(0,\infty)$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works. |
| title | Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.02466 |