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 R-d and dimension d >= 3 the critical spaces for the initial bacteria density and the chemical gradient are respectively L-a(R-d), a > d/2, and L-d (R-d). 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. (C) 2007 Elsevier Ltd. All rights reserved.
asymptotic decay chemotaxis hypercontractivity parabolic systems global weak solutions