Large time behaviour of higher dimensional logarithmic diffusion equation

Abstract: Let $n\ge 3$ and $\psi_{\lambda_0}$ be the radially symmetric solution of $\Delta\log\psi+2\beta\psi+\beta x\cdot\nabla\psi=0$ in $R^n$, $\psi(0)=\lambda_0$, for some constants $\lambda_0>0$, $\beta>0$. Suppose $u_0\ge 0$ satisfies $u_0-\psi_{\lambda_0}\in L^1(R^n)$ and $u_0(x)\approx\frac{2(n-2)}{\beta}\frac{\log |x|}{|x|^2}$ as $|x|\to\infty$. We prove that the rescaled solution $\widetilde{u}(x,t)=e^{2\beta t}u(e^{\beta t}x,t)$ of the maximal global solution $u$ of the equation $u_t=\Delta\log u$ in $R^n\times (0,\infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subset of $R^n$ and in $L^1(R^n)$ to $\psi_{\lambda_0}$ as $t\to\infty$. Moreover $|\widetilde{u}(\cdot,t)-\psi_{\lambda_0}|{L^1(R^n)} \le e^{-(n-2)\beta t}|u_0-\psi{\lambda_0}|_{L^1(R^n)}$ for all $t\ge 0$.