Séminaire EDP-Analyse ICJ

Multi-layer patterns for the stationary Keller-Segel system

par Denis Bonheure (Université Libre de Bruxelles)

Fokko Du Cloux (Université Claude Bernard Lyon 1 - Campus de la Doua, Bâtiment Braconnier)

Fokko Du Cloux

Université Claude Bernard Lyon 1 - Campus de la Doua, Bâtiment Braconnier

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} -\varepsilon^2 \Delta v+v=v^p, \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} -\varepsilon^2 \Delta v+v=\lambda e^v, \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 (Roman), Susanna Teracinni (Torino), Benedetta Noris (Amiens), Christophe Troestler (Mons), Jean-Baptiste Casteras (Bruxelles) and Carlos Roman (UPMC).
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