Hokkaido Mathematical Journal

Selfsimilar solutions of the porous medium equation without sign restriction

Claus DOHMEN

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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

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1381413101

Digital Object Identifier
doi:10.14492/hokmj/1381413101

Mathematical Reviews number (MathSciNet)
MR1299639

Zentralblatt MATH identifier
0818.35051

Subjects
Primary: 35K65: Degenerate parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations

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


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