Complete classification of self-similar solutions for singular polytropic filtration equations
This article concerns the complete classification of self-similar solutions to the singular polytropic filtration equation. We establish the existence and uniqueness of self-similar solutions of the form $u(x,t)=(\beta t)^{-\alpha/\beta}w((\beta t)^{-\frac{1}{\beta}}|x|)$, and the regularity or sing...
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| Format: | Article |
| Language: | English |
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Texas State University
2025-06-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2025/63/abstr.html |
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| author | Yanzhi Zheng Jingxue Yin Shanming Ji |
| author_facet | Yanzhi Zheng Jingxue Yin Shanming Ji |
| author_sort | Yanzhi Zheng |
| collection | DOAJ |
| description | This article concerns the complete classification of self-similar solutions to the
singular polytropic filtration equation.
We establish the existence and uniqueness of self-similar solutions of the form
$u(x,t)=(\beta t)^{-\alpha/\beta}w((\beta t)^{-\frac{1}{\beta}}|x|)$,
and the regularity or singularity at $x=0$, with $\alpha,\beta\in\mathbb{R}$ and
$\beta=p-\alpha(1-mp+m)$.
The asymptotic behaviors of the solutions near 0 orinfinity are also described.
Specifically, when $\beta<0$, there always exist blow up solutions or oscillatory
solutions. When $\beta>0$, oscillatory solutions appear if $\alpha>N$, $0<m<1$ and $1<p<2$.
The main technical issue for the proof is to overcome the difficulty arising from the
doubly nonlinear non-Newtonian polytropic filtration diffusion
$\text{div}({|\nabla u^m|}^{p-2} \nabla u^m)$. |
| format | Article |
| id | doaj-art-e9e488dbfbea46ca82b28b533a6b92eb |
| institution | Kabale University |
| issn | 1072-6691 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj-art-e9e488dbfbea46ca82b28b533a6b92eb2025-08-20T03:43:40ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912025-06-01202563,132Complete classification of self-similar solutions for singular polytropic filtration equationsYanzhi Zheng0Jingxue Yin1Shanming Ji2 South China Normal Univ., Guangzhou, Guangdong, China South China Normal Univ., Guangzhou, Guangdong, China South China Normal Univ., Guangzhou, Guangdong, China This article concerns the complete classification of self-similar solutions to the singular polytropic filtration equation. We establish the existence and uniqueness of self-similar solutions of the form $u(x,t)=(\beta t)^{-\alpha/\beta}w((\beta t)^{-\frac{1}{\beta}}|x|)$, and the regularity or singularity at $x=0$, with $\alpha,\beta\in\mathbb{R}$ and $\beta=p-\alpha(1-mp+m)$. The asymptotic behaviors of the solutions near 0 orinfinity are also described. Specifically, when $\beta<0$, there always exist blow up solutions or oscillatory solutions. When $\beta>0$, oscillatory solutions appear if $\alpha>N$, $0<m<1$ and $1<p<2$. The main technical issue for the proof is to overcome the difficulty arising from the doubly nonlinear non-Newtonian polytropic filtration diffusion $\text{div}({|\nabla u^m|}^{p-2} \nabla u^m)$.http://ejde.math.txstate.edu/Volumes/2025/63/abstr.htmlpolytropic filtration equationself-similar solutionssingularity phase plane analysisasymptotic behavior |
| spellingShingle | Yanzhi Zheng Jingxue Yin Shanming Ji Complete classification of self-similar solutions for singular polytropic filtration equations Electronic Journal of Differential Equations polytropic filtration equation self-similar solutions singularity phase plane analysis asymptotic behavior |
| title | Complete classification of self-similar solutions for singular polytropic filtration equations |
| title_full | Complete classification of self-similar solutions for singular polytropic filtration equations |
| title_fullStr | Complete classification of self-similar solutions for singular polytropic filtration equations |
| title_full_unstemmed | Complete classification of self-similar solutions for singular polytropic filtration equations |
| title_short | Complete classification of self-similar solutions for singular polytropic filtration equations |
| title_sort | complete classification of self similar solutions for singular polytropic filtration equations |
| topic | polytropic filtration equation self-similar solutions singularity phase plane analysis asymptotic behavior |
| url | http://ejde.math.txstate.edu/Volumes/2025/63/abstr.html |
| work_keys_str_mv | AT yanzhizheng completeclassificationofselfsimilarsolutionsforsingularpolytropicfiltrationequations AT jingxueyin completeclassificationofselfsimilarsolutionsforsingularpolytropicfiltrationequations AT shanmingji completeclassificationofselfsimilarsolutionsforsingularpolytropicfiltrationequations |