View abstract

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).

View abstract PDF