Two-dimensional chemotaxis models with fractional diffusion

✍ Scribed by Piotr Biler; Gang Wu

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 143 KB
• Volume: 32
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1036

✦ Synopsis

Abstract

In this paper we study the Cauchy problem for the fractional diffusion equation u~t~ + (−Δ)^α/2^u=∇·(u∇(Δ^−1^u)), generalizing the Keller–Segel model of chemotaxis, for the initial data u~0~ in critical Besov spaces Ḃ(ℝ^2^) with r∈[1, ∞], where 1<α<2. Making use of some estimates of the linear dissipative equation in the frame of mixed time–space spaces, the Chemin ‘mono‐norm method,’ Fourier localization technique and the Littlewood–Paley theory, we obtain a local well‐posedness result. We also consider analogous ‘doubly parabolic’ models. Copyright © 2008 John Wiley & Sons, Ltd.