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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2025-06-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2025/63/abstr.html |
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| Summary: | 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
$\text{div}({|\nabla u^m|}^{p-2} \nabla u^m)$. |
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| ISSN: | 1072-6691 |