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, $$\left\{ \begin{array}{l} \partial_t u=\partial_{xx} \log u\quad \mbox{in}\quad \left[-l,l\right]\times \left(0, \infty\right)\\ \displaystyle \partial_x u\left(\pm l, t\right)=\pm 2\gamma u^{p}\left(\pm l, t\right), \end{array} \right.$$ 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 $\gamma > 0$, $p\leq \frac{3}{2}$ 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 $\gamma<0$, $p\geq \frac{3}{2}$, solutions to the logarithmic diffusion equation with initial data $u_0$ are global and blow-down in infinite time, but if $p\leq 1$ 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.