On very singular self-similar solutions for the porous media equation with absorption
Consider the equation \[ w_\tau =\Delta (w^m)-w^p \] in \(\mathbb {R}^n\times \mathbb {R}^+\), where \(p>m>1\). It is shown that this equation admits a very singular self-similar solution~\(w\) of the form \[ w(x,t)=\tau ^{-a/b}W\bigl (\tau ^{-1/b}|x|\bigr), \] where \(a=2/(p-m)\), \(b=2(p-1)/(p-m)\), and \(W\)~is a non-negative solution of the problem \[ (W^m)''+\frac {n-1}{t}\bigl (W^m\bigr)'+\frac {1}{b}tW'+ \frac {a}{b}W-W^p=0,\qquad W'(0)=0,\quad \lim _{t\to +\infty}t^aW(t)=0, \] if and only if \(p<m+(2/n)\). Further the initial value problem \[ w_\tau =\Delta (w^m)\quad \text{in \(\mathbb {R}^n\times \mathbb {R}^+\),}\qquad w(x,0)=C|x|^{-a}\quad \text{in \(\mathbb {R}^n\setminus \{0\}\),} \] where \(m>1\), \(a,C>0\), is studied, and conditions for existence of a self-similar solution (of a specified form) are given.
scientific article; zbMATH DE number 1450172