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 $\tilde{u}(x,t)=e^{2\beta t}u(e^{\beta t}x,t)$ 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 $\Vert \tilde{u}(\cdot ,t)-{\psi }_{{\lambda }_0}{\Vert }_{L^1(R^n)}\le e^{-(n-2)\beta t}\Vert u_0-{\psi }_{{\lambda }_0}{\Vert }_{L^1(R^n)}$ for all $t\ge 0$.

12 pages