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arXiv 2103.00240math.APmath.DG
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具有非线性Robin边界条件的一维对数扩散方程

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

Jean Cortissoz, César Reyes

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摘要：

研究了一维对数扩散方程的非线性Robin边界条件，即\[ \left\{ \begin{array}{l} \partial_t u=\partial_{xx} \log u\quad \mbox{in}\quad \left[-l,l\right]\times \left(0, \infty\right)\\ \displaystyle \partial_x u\left(\pm l, t\right)=\pm 2γu^{p}\left(\pm l, t\right), \end{array} \right. \]，其中\[ \left\{ \begin{array}{l} \partial_t u=\partial_{xx} \log u\quad \mbox{in}\quad \left[-l,l\right]\times \left(0, \infty\right)\\ \displaystyle \partial_x u\left(\pm l, t\right)=\pm 2γu^{p}\left(\pm l, t\right), \end{array} \right. \]为常数.设$u_0>0$是定义在$\left[-l,l\right]$上的光滑函数，且满足相容条件$$\partial_x \log u_0\left(\pm l\right)= \pm 2γu_0^{p-1}\left(\pm l\right).$$.我们证明了对于$γ> 0$，$p\leq \frac{3}{2}$，上述对数扩散方程的解是全局的，且在无限时间内爆破，而对于$p>2$，存在有限时间爆破.此外，我们还证明了在$γ<0$，$p\geq \frac{3}{2}$的情况下，以$u_0$为初始值的对数扩散方程的解是全局的，并且在无限时间内是blow-down的，但如果是$p\leq 1$，则存在有限时间blow-down。对于上面提到的一些情况，以及一些特殊的例子族，我们提供了爆破和爆破率。我们的方法是部分基于研究的Ricci流的圆柱赋予$\mathbb{S}^1$对称度量。然后，我们把我们的想法完整的循环证明一个新的长时间存在的结果的Ricci流的圆柱没有任何对称性的假设。最后，我们给出了圆盘上对数扩散方程解的一个爆破结果。

英文摘要：

In this paper we investigate the one dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely, \[ \left\{ \begin{array}{l} \partial_t u=\partial_{xx} \log u\quad \mbox{in}\quad \left[-l,l\right]\times \left(0, \infty\right)\\ \displaystyle \partial_x u\left(\pm l, t\right)=\pm 2γu^{p}\left(\pm l, t\right), \end{array} \right. \] where $γ$ is a constant. Let $u_0>0$ be a smooth function defined on $\left[-l,l\right]$, and which satisfies the compatibility condition $$\partial_x \log u_0\left(\pm l\right)= \pm 2γu_0^{p-1}\left(\pm l\right).$$ We show that for $γ> 0$, $p\leq \frac{3}{2}$ 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 $γ<0$, $p\geq \frac{3}{2}$, 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.

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