CRM: Centro De Giorgi
logo sns
New trends in Partial Differential Equations

Multi-layer solutions for the stationary Keller-Segel system

speaker: Denis Bonheure (ULB)

abstract: The stationary Keller-Segel system, which is a classical model for chemotaxis, leads (with some specific choice in the model) to the Lin-Ni-Takagi equation \begin{equation}\label{lnt} \begin{cases} -\varepsilon2 \Delta v+v=vp, \quad v>0 \quad & \text{in }\Omega,\\ \partial\nu v=0 &\text{on } \partial \Omega, \end{cases} \end{equation} and the Keller-Segel equation \begin{equation}\label{ks} \begin{cases} -\varepsilon2 \Delta v+v=\lambda ev, \quad v>0 \quad & \text{in }\Omega,\\ \partial\nu v=0 &\text{on } \partial \Omega. \end{cases} \end{equation} In this talk, I will show how to build radial multi-layer solutions, assuming \(\Omega\) is a ball, of those equations using either bifurcations, the Lyapunov-Schmidt method or a variational gluing method. It is a remarkable fact that even if those solutions are obtained in an asymptotic regime (\(p\to \infty\) or \(\lambda\to 0\)), the layers do not accumulate on the boundary of the ball. Instead they solve an optimal partition problem.

The talk is based on several joint works with Massimo Grossi, Susanna Teracinni, Benedetta Noris, Christophe Troestler and Jean-Baptiste Casteras.

Wed 5 Oct, 11:00 - 11:45, Sala Stemmi
<< Go back