Asymptotic decay for the solutions of the parabolic–parabolic Keller–Segel chemotaxis system in critical spaces

We consider the classical parabolic–parabolic Keller–Segel system describing chemotaxis, i.e., when both the evolution of the biological population and the chemoattractant concentration are described by a parabolic equation. We prove that when the equation is set in the whole space RdRd and dimension d≥3d≥3 the critical spaces for the initial bacteria density and the chemical gradient are respectively La(Rd)La(Rd), a>d/2a>d/2, and Ld(Rd)Ld(Rd). For 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.