Yanzhi Zheng, Jingxue Yin, Shanming Ji
Abstract:
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
\(u(x,t)=(\beta t)^{-\alpha/\beta}w((\beta t)^{-1/\beta} |x|)\),
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
\( \hbox{div}({|\nabla u^m|}^{p-2} \nabla u^m)\).
Submitted April 29, 2025. Published June 26, 2025.
Math Subject Classifications: 35K67, 35C06, 35K92, 35B40.
Key Words: Polytropic filtration equation; self-similar solutions; singularity; phase plane analysis; asymptotic behavior.
DOI: 10.58997/ejde.2025.63
Show me the PDF file (2877 KB), TEX file for this article.
 |
Yanzhi Zheng School of Mathematical Sciences South China Normal University Guangzhou, Guangdong, 510631, China email: zhengyanzhi51@163.com |
|---|---|
 |
Jingxue Yin School of Mathematical Sciences South China Normal University Guangzhou, Guangdong, 510631, China email: yjx@scnu.edu.cn |
 |
Shanming Ji School of Mathematics South China University of Technology Guangzhou, Guangdong, 510641, China email: jism@scut.edu.cn |
Return to the EJDE web page