Complete classification of self-similar solutions for singular polytropic filtration equations - Yanzhi Zheng; Jingxue Yin; Shanming Ji (Electronic Journal of Differential 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 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 $\text{div}({|\nabla u^m|}^{p-2} \nabla u^m)$.