Large time behaviour of higher dimensional logarithmic diffusion equation

Abstract

Let $n\ge 3$ and $ψ_{λ_0}$ be the radially symmetric solution of $Δ\logψ+2βψ+βx\cdot\nablaψ=0$ in $R^n$, $ψ(0)=λ_0$, for some constants $λ_0>0$, $β>0$. Suppose $u_0\ge 0$ satisfies $u_0-ψ_{λ_0}\in L^1(R^n)$ and $u_0(x)\approx\frac{2(n-2)}β\frac{\log |x|}{|x|^2}$ as $|x|\to\infty$. We prove that the rescaled solution $\widetilde{u}(x,t)=e^{2βt}u(e^{βt}x,t)$ of the maximal global solution $u$ of the equation $u_t=Δ\log u$ 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 $ψ_{λ_0}$ as $t\to\infty$. Moreover $\|\widetilde{u}(\cdot,t)-ψ_{λ_0}\|_{L^1(R^n)} \le e^{-(n-2)βt}\|u_0-ψ_{λ_0}\|_{L^1(R^n)}$ for all $t\ge 0$.

12 pages