We study the Keller-Segel system in Rd when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies n0 ∈ La ( Rd ), a > d / 2, and that the chemoattractant concentration satisfies ∇ c0 ∈ Ld ( Rd ). In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type. To cite this article: L. Corrias, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 342 (2006). © 2006 Académie des sciences.
Corrias, L., & Perthame, B. (2006). Critical space for the parabolic-parabolic Keller-Segel model in Rd. Comptes Rendus Mathematique, 342(10), 745–750. https://doi.org/10.1016/j.crma.2006.03.008