Two‐dimensional chemotaxis models with fractional diffusion
Piotr Biler, Gang Wu
Source abstract
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.
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