The one-dimensional Keller-Segel model with fractional diffusion of cells

Nikolaos Bournaveas (1) , Vincent Calvez (2, 3)

1 School of Mathematics - University of Edinburgh
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 NUMED - Numerical Medicine

We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent $0<\alpha\leq 2$. We prove some features related to the classical two-dimensional Keller-Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when $\alpha<1$ and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for $\alpha\leq1$ if the initial density is small enough in the sense of the $L^{1/\alpha}$ norm.

Mots clés: Self-organization; chemotaxis; fractional diffusion; global existence; blow-up.; blow-up

Domaines: Equations aux dérivées partielles [math.AP]

Nikolaos Bournaveas, Vincent Calvez. The one-dimensional Keller-Segel model with fractional diffusion of cells. 2009. ⟨hal-00398629⟩