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2005 Classification of radially symmetric self-similar solutions of $u_t=\Delta log u$ in higher dimensions
Shu-Yu Hsu
Differential Integral Equations 18(10): 1175-1192 (2005).

Abstract

We give a complete classification of radially symmetric self-similar solutions of the equation $u_t=\Delta\log u$, $u>0$, in higher dimensions. For any $n\ge 2$, $\eta>0$, $\alpha$, $\beta\in \mathbb {R}$, we prove that there exists a radially symmetric solution for the corresponding elliptic equation $\Delta\log v+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $ \mathbb {R}^n$, $v(0)=\eta$, if and only if either $\alpha\ge 0$ or $\beta>0$. For $n\ge 3$, we prove that $\lim_{r\to\infty}r^2v(r) =2(n-2)/(\alpha -2\beta)$ if $\alpha>\max (2\beta,0)$ and $\lim_{r\to\infty}r^2v(r)/\log r=2(n-2)/\beta$ if $\alpha=2\beta>0$. For $n\ge 2$ and $2\beta>\max (\alpha ,0)$, we prove that $\lim_{r\to\infty}r^{\alpha/\beta}v(r)=A$ for some constant $A>0$.

Citation

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Shu-Yu Hsu. "Classification of radially symmetric self-similar solutions of $u_t=\Delta log u$ in higher dimensions." Differential Integral Equations 18 (10) 1175 - 1192, 2005.

Information

Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35150
MathSciNet: MR2162628

Subjects:
Primary: 34B40
Secondary: 34B18 , 34C11 , 35K55

Rights: Copyright © 2005 Khayyam Publishing, Inc.

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Vol.18 • No. 10 • 2005
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