Critical space for the parabolic-parabolic Keller–Segel model in Rd

Abstract We study the Keller–Segel system in R d 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 n 0 ∈ L a ( R d ) , a > d / 2 , and that the chemoattractant concentration satisfies ∇ c 0 ∈ L d ( R d ) . 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).