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

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 $A>0$.