Hokkaido Mathematical Journal

Selfsimilar solutions of the porous medium equation without sign restriction

Claus DOHMEN

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.

Article information

Source
Hokkaido Math. J., Volume 23, Number 3 (1994), 475-505.

Dates
First available in Project Euclid: 10 October 2013

https://projecteuclid.org/euclid.hokmj/1381413101

Digital Object Identifier
doi:10.14492/hokmj/1381413101

Mathematical Reviews number (MathSciNet)
MR1299639

Zentralblatt MATH identifier
0818.35051

Citation

DOHMEN, Claus. Selfsimilar solutions of the porous medium equation without sign restriction. Hokkaido Math. J. 23 (1994), no. 3, 475--505. doi:10.14492/hokmj/1381413101. https://projecteuclid.org/euclid.hokmj/1381413101