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

✍ Scribed by Lucilla Corrias; Benoǐt Perthame

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 273 KB
• Volume: 47
• Category: Article
• ISSN: 0895-7177
• DOI: 10.1016/j.mcm.2007.06.005

✦ Synopsis

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.