Session S39 - Differential Equations and Geometric Structures

Monday, July 19, 19:00 ~ 19:50 UTC-3

On the logarithmic diffusion equation

Jean Carlos Cortissoz

Universidad de los Andes, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will consider the boundary value problem \[ \left\{ \begin{array}{l} \partial_t u=\partial_{xx} \log u\quad \mbox{in}\quad \left[-l,l\right]\times \left(0, \infty\right)\\ \displaystyle \partial_x u\left(\pm l, t\right)=\pm 2\gamma u^{p}\left(\pm l, t\right), \end{array} \right. \] 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 $$\partial_x \log u_0\left(\pm l\right)= \pm 2\gamma u_0^{p-1}\left(\pm l\right).$$

This equation is closely related to the Ricci flow on a cylinder. We will use this relation to show some results on the behavior of solutions to this equation . For instance, depending on $p$, these solutions can be global or not, and depending on the sign of $\gamma$ there could be blow-up or blow-down (quenching) in finite or infinte time (in the case the solution is global). This is joint work with C\'esar Reyes.

Joint work with César Reyes (Universidad Manuela Beltrán).

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