Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 63, pp. 1-32.

Title: Complete classification of self-similar solutions for singular polytropic filtration equations

Authors: Yanzhi Zheng (South China Normal Univ., Guangzhou, Guangdong, China)
         Jingxue Yin (South China Normal Univ., Guangzhou, Guangdong, China)
         Shanming Ji (South China Normal Univ., Guangzhou, Guangdong, China)

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
$u(x,t)=(\beta t)^{-\alpha/\beta}w((\beta t)^{-\frac{1}{\beta}}|x|)$,
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
$\text{div}({|\nabla u^m|}^{p-2} \nabla u^m)$.

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.