Differential and Integral Equations

Classification of radially symmetric self-similar solutions of $u_t=\Delta log u$ in higher dimensions

Shu-Yu Hsu

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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 $A>0$.

Article information

Differential Integral Equations, Volume 18, Number 10 (2005), 1175-1192.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B40: Boundary value problems on infinite intervals
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 34C11: Growth, boundedness 35K55: Nonlinear parabolic equations


Hsu, Shu-Yu. Classification of radially symmetric self-similar solutions of $u_t=\Delta log u$ in higher dimensions. Differential Integral Equations 18 (2005), no. 10, 1175--1192. https://projecteuclid.org/euclid.die/1356060110

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