A very singular solution for the porous media equation \(u_ t=\Delta(u^ m)-u^ p\) when \(0<m<1\)
Of concern is the existence of a very singular selfsimilar solution of the porous media equation with absorption \[ \partial_t u= \Delta(u^m)-u^p \quad\text{on}\quad {\mathbb{R}}^n\times{\mathbb{R}}_+ \tag{PMA} \] in the fast diffusion case \(0<m<1\), if \(p>1\). Recall that a nonnegative solution \(w\) of (PMA) is called very singular, if \(w\) is smooth on \(\bigl({\mathbb{R}}^n \times{\mathbb{R}}_+\bigr)\setminus\{(0,0)\}\) and \(\displaystyle\lim_{\tau\to 0^+}\int_{\left|x \right|<\tau} w(y,\tau)dy=\infty\). Assuming that \(m>(1-(2/n))^+\) and \(p<m+(2/n)\) the author establishes the existence of a very singular solution \(w\) of (PMA) of the form \(w(x,\tau)=\tau^{a/b}W(\tau^{-1/b}\left|x \right|)\), where \(a:=2/(p-m)\), \(b:=2{{ p-1}\over{p-m}}\) and W is a nontrivial nonnegative solution of \[ (W^m)''(s)+{\textstyle {{{n-1}\over s}(W^m)'(s)+{s\over b}}W''(s)+{a\over b}W(s) -W(s)^p=0} \] (\( s\in (0,\infty]\)) satisfying \(W'(0)=0\) and \(\lim_{s\to\infty}s^aW(s)=0\). If the conditions for \(m\) or \(p\) fail, the author proves that all solutions of the ordinary differential equation with \(W'(0)=0\) fulfil \(\lim_{s\to\infty}s^aW(s)>0\). The proofs rely on shooting arguments. The existence of very singular solutions for the heat equation with absorption goes back to \textit{H. Brezis}, \textit{L. A. Peletier} and \textit{D. Terman} [Arch. Ration. Mech. Anal. 95, 185-209 (1985; Zbl 0627.35046)], whereas the slow diffusion case \(m>1\) has been considered by the author for \(m<p<m+2/n\).
scientific article; zbMATH DE number 1000167