Large time behaviour of higher dimensional logarithmic diffusion equation

TL;DR

This paper studies the long-term behavior of solutions to a higher-dimensional logarithmic diffusion equation, proving convergence to a radially symmetric stationary solution under specific initial conditions and providing explicit decay rates in L^1 norm.

Contribution

It establishes the asymptotic convergence of rescaled solutions to a radially symmetric stationary solution for the higher-dimensional logarithmic diffusion equation, with explicit decay estimates.

Findings

Rescaled solutions converge uniformly on compact sets and in L^1 to the stationary solution.

The convergence rate in L^1 norm is exponentially fast, with decay depending on the dimension and parameters.

The paper characterizes the asymptotic profile for solutions with specific initial decay at infinity.

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 $t\ge 0$.