On the one dimensional Logarithmic diffusion equation with nonlinear Robin boundary conditions

In this paper we investigate the one dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely, $$\{\begin{array}{l}{\partial }_{t}u={\partial }_{xx}\log u\text{mbox}in\left[-l,l\right]\times \left(0,\infty \right)\\ {\partial }_{x}u\left(\pm l,t\right)=\pm 2\gamma u^p\left(\pm l,t\right),\end{array}$$ where $\gamma$ is a constant. Let $u_0>0$ be a smooth function defined on $\left[-l,l\right]$, and which satisfies the compatibility condition $${\partial }_x\log u_0\left(\pm l\right)=\pm 2\gamma u_0^{p-1}\left(\pm l\right).$$ We show that for $\gamma >0$, $p\le \frac{3}{2}$ solutions to the logarithmic diffusion equation above with initial data $u_0$ are global and blow-up in infinite time, and for $p>2$ there is finite time blow-up. Also, we show that in the case of $\gamma <0$, $p\ge \frac{3}{2}$, solutions to the logarithmic diffusion equation with initial data $u_0$ are global and blow-down in infinite time, but if $p\le 1$ there is finite time blow-down. For some of the cases mentioned above, and some particular families of examples, we provide blow-up and blow-down rates. Our approach is partly based on studying the Ricci flow on a cylinder endowed with a ${\mathbb{S}}^1$-symmetric metric. Then, we bring our ideas full circle by proving a new long time existence result for the Ricci flow on a cylinder without any symmetry assumption. Finally, we show a blow-down result for the logarithmic diffusion equation on a disc.