Selfsimilar solutions of the porous medium equation without sign restriction

Abstract

We consider radially symmetric selfsimilar solutions $u(x,t)=t^{-a}U(|x|t^{-\beta})$ of the porous medium equation $u_{t}-\Delta(|u|^{m-1}u)=0$. If $m\in (\frac{(N-2)_{+}}{N} , 1 )$, we show that the resulting ODE allows global solutions with rapid decay for a sequence of parameters $k=\alpha/\beta$, denoted by $\{k_{i}^{g}(m, N)\}_{i\in N}\subset {\mathbf R}^{+}$. The corresponding solution $U_{i}$ has exactly $(i-1)$ simple zeroes in ${\mathbf R}^{+}$ This case was left open by previous papers, where the result for the degenerate case was given. Besides the existence result in the singular case $m<1$ for arbitrary space dimension N we prove continuity of the $k_{i}^{g}(., N)$ at functions of m in $(\frac{(N-2)_{+}}{N}, 1)$. In one space dimension there also exist antisymmetric solutions with rapid decay for certain values $\{k_{i}^{u}(m)\}_{i\in N}$ . We show that these values as well as the $k_{i}^{g}(., 1)$ are continuous functions of m in ${\mathbf R}^{+}$ and identify their limits marrow\infty with compactly supported solutions of a limit problem.