We study the Keller-Segel system in Rdwhen 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.
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